course Mth 151 I included much more detail in my responses this time, as requested. I hope this is more in line with what you are looking for. h?S??????????assignment #006
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19:04:12 `q001. Note that there are 6 questions in this assignment. Find the likely next element of the sequence 1, 2, 4, 7, 11, ... .
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RESPONSE --> the next element would most likely be16. Each number is being added with the next ""counting"" number is the sequence. By this I mean we can see that 1+1= 2, 2+2= 4, 4+3=7, 7+4=11, 11+5= 16. confidence assessment: 3
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19:04:44 The difference between 1 and 2 is 1; between 2 and 4 is 2; between 4 and 7 is 3; between 7 and 11 is 4. So we expect that the next difference will be 5, which will make the next element 11 + 5 = 16.
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RESPONSE --> No critique is needed because this is what I was trying to explain. self critique assessment: 3
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19:07:07 `q002. Find the likely next two elements of the sequence 1, 2, 4, 8, 15, 26, ... .
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RESPONSE --> I am not able to deduct the next two elements in this problem by using what is provided. I cannot find any addition, subtraction, or multiplication relation between the elements. confidence assessment: 1
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19:13:10 The difference between 1 and 2 is 1; the difference between 2 and 4 is 2, the difference between 4 and 8 is 4; the difference between 8 and 15 is 7; the difference between 15 and 26 is 11. The differences form the sequence 1, 2, 4, 7, 11, ... . As seen in the preceding problem the differences of this sequence are 1, 2, 3, 4, ... . We would expect the next two differences of this last sequence to be 5 and 6, which would extend the sequence 1, 2, 4, 7, 11, ... to 1, 2, 4, 7, 11, 16, 22, ... . If this is the continuation of the sequence of differences for the original sequence 1, 2, 4, 8, 15, 26, ... then the next two differences of this sequence would be 16 , giving us 26 + 16 = 42 as the next element, and 22, giving us 42 + 26 = 68 as the next element. So the original sequence would continue as 1, 2, 4, 8, 15, 26, 42, 68, ... .
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RESPONSE --> I understand where they get the differences, but what I am not understanding is where the extra numbers like 15 and 26 come into the picture?
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19:14:44 `q003. What would be the likely next element in the sequence 1, 2, 4, 8, ... . It is understood that while this sequence starts off the same as that in the preceding exercise, it is not the same. The next element is not 15, and the pattern of the sequence is different than the pattern of the preceding.
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RESPONSE --> I think the next likely element would be 16. If you work backwards, you can divde 16 by two and get 8. 8/2=4, 4/2=2, and 2/2= 1. If taken in this order, it makes sense for 16 to be the next element in the problem. confidence assessment: 3
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19:15:11 One obvious pattern for this sequence is that each number is doubled to get the next. If this pattern continues then the sequence would continue by doubling 8 to get 16. The sequence would therefore be 1, 2, 4, 8, 16, ... .
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RESPONSE --> no critique is needed, this is the answer I got for the problem. self critique assessment: 3
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19:27:51 `q004. There are two important types of patterns for sequences, one being the pattern defined by the differences between the numbers of the sequence, the other being the pattern defined by the ratios of the numbers of the sequence. In the preceding sequence 1, 2, 4, 8, 16, ..., the ratios were 2/1 = 2; 4/2 = 2; 8/4 = 2; 16/8 = 2. The sequence of ratios for 1, 2, 4, 8, 16, ..., is thus 2, 2, 2, 2, a constant sequence. Find the sequence of ratios for the sequence 32, 48, 72, 108, ... , and use your result to estimate the next number and sequence.
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RESPONSE --> I think the next number in the order would be 160. When you subtract the bigger number in the sequence from the smaller one, you get certain numbers. 48-32=16 72=48=24 108-72= 36 160-108=52 Each number in the sequence is mult. by four as the sequence. confidence assessment: 2
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19:28:53 The ratios are 48/32 = 1.5; 72 / 48 = 1.5; 108/72 = 1.5, so the sequence of ratios is 1.5, 1.5, 1.5, 1.5, ... . The next number the sequence should probably therefore be 108 * 1.5 = 162.
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RESPONSE --> I understand how they did this after seeing the answer. For some reason I was trying to make it more difficult in my head and was adding and them multiplying instead of dividing to get the ratios. self critique assessment: 2
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19:33:51 `q005. Find the sequence of ratios for the sequence 1, 2, 3, 5, 8, 13, 21... , and estimate the next element of the sequence.
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RESPONSE --> I think the next number is 30. with each number they are adding two numbers on the other end. 8+5=13, so 13+7 =21, so 21+9 would be 30 confidence assessment: 2
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19:37:08 The ratios are 2/1 = 2; 3/2 = 1.5; 5/3 = 1.66...; 8/5 = 1.60; 13/8 = 1.625; 21/13 = 1.615. The sequence of ratios is 2, 1.5, 1.66..., 1.625, 1.615, ... . We see that each number in the sequence lies between the two numbers that precede it -- 1.66... lies between 2 and 1.5; 1.60 lies between 1.5 and 1.66...; 1.625 lies between 1.66... and 1.60; 1.615 lies between 1.60 and 1.625. We also see that the numbers in the sequence alternate between being greater than the preceding number and less than the preceding number, so that the intervals between the numbers get smaller and smaller. So we expect that the next number in the sequence of ratios will be between 1.615 and 1.625, and if we pay careful attention to the pattern we expect the next number to be closer to 1.615 than to 1.625. We might therefore estimate that the next ratio would be about 1.618. We would therefore get 1.618 * 21 = 33.98 for the next number in the original sequence. However, since the numbers in the sequence are all whole numbers, we round our estimate up to 34. Our conjecture is that the sequence continues with 1, 2, 3, 5, 8, 13, 21, 34, ... .
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RESPONSE --> When I did the division for this problem, I did notice that all of the ""1.6..."" relation between, but found no way I could understand to use that to find the next whole number that might do the same. In the answer they broke it down farther than I was able to do and actually got the dec. number and rounded to the nearest whole number of 34. I apparently didnt pay enough attention to the dec. pattern when I did the divison of each numbers. self critique assessment: 2
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19:40:23 `q006. Without using ratios, can you find a pattern to the sequence 1, 1, 2, 3, 5, 8, 13, 21, 34, ..., and continue the sequence for three more numbers?
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RESPONSE --> 1+1=2 2+3=5 3+5=8 8+5=13 13+8=21 21+13=34 34+21=55 55+21=76 confidence assessment: 3
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19:41:11 The pattern is that each element from the third on is the sum of the two elements that precede it. That is, 1+1=2, 2+1=3; 3+2=5; 5+3=8; 8+5=13; 13+8=21; 21+13=34; . The next three elements would therefore e 34+21=55; 55+34=89; 89+55=144. . The sequence is seen to be 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ... .
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RESPONSE --> This is pretty much what I was gettting at, but somehow I got the simple addition mixed up when I was doing it in the submission box. self critique assessment: 2
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~L?????????? Student Name: assignment #007
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19:45:37 `q001. Note that there are 7 questions in this assignment. Sketch three points A, B and C forming an equilateral triangle on a piece of paper, with point A at the lower left-hand corner, point B at the lower right-hand corner and point C at the top. Sketch the segments AB and AC. Now double the lengths of AB and AC, and place a point at each of the endpoints of these segments. Connect these new endpoints to form a new equilateral triangle. Two sides of this triangle will have three points marked while the new side will only have its two endpoints marked. Fix that by marking that middle point, so all three sides of your new triangle are marked the same. How many marked points were there in the original triangle, and how many are there in the new triangle?
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RESPONSE --> 3 in the first and 4 in the second
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19:46:00 The original triangle had the three points A, B and C. When you extended the two sides you marked the new endpoints, then you marked the point in the middle of the third side. So you've got 6 points marked. Click on 'Next Picture' to see the construction. The original points A, B and C are shown in red. The line segments from A to B and from A to C have been extended in green and points marked at the ends of these segments. The new endpoints have been connected to form the third side of a larger triangle, and an equally spaced point has been constructed at the midpoint of that side. Your figure should contain the three original points, plus the three points added when the new side was completed.
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RESPONSE -->
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19:46:48 `q002. Extend the two sides that meet at A by distances equal to the distance original lengths AC and AB and mark the endpoints of the newly extended segments. Each of the newly extended sides will have 4 marked points. Now connect the new endpoints to form a new right triangle. Mark points along the new side at the same intervals that occur on the other two sides. How many marked points are on your new triangle, and how many in the whole figure?
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RESPONSE --> 3 on the new triangle and six total marked points
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19:46:56 You added the two new endpoints when you extended the sides. You then should have marked two new points on the new third side, so that each side contains 4 points including its endpoints. Your figure will now contain 10 marked points.
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RESPONSE -->
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19:48:21 `q003. Continue the process for another step-extend each side by a distance equal to the original point-to-point distance. How many points do you have in the new triangle?
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RESPONSE --> you would have 13 in the new triangle I think. If you had the 10 and then the new segment by the org. triangle, 10 +3=13 points
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19:49:01 You will add an endpoint to each newly extended side, so each of the new sides will contain 5 points. You will then have to add 3 equally spaced points to the new side, giving you a total of 13 points on the new triangle. In addition there are two marked points inside the triangle, for a total of 15 points. Click on 'Next Picture' to see the construction. The line segments along two sides of the triangle have again been extended and points marked at the ends of these segments. The new endpoints have been connected to form the third side of a larger triangle, and equally spaced points have been constructed along that side.
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RESPONSE --> I'm not exactly sure the program is on the correct problems because none of this looks like the other assignments
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19:49:24 `q004. Continue the process for one more step. How many points do you have in the new triangle?
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RESPONSE --> 21 in the new triangle
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19:50:11 You will add an endpoint to each newly extended side, so each of the new sides will contain 6 points. You will then have to add 4 equally spaced points to the new side, giving you a total of 15 points on the new triangle. There are also 5 marked points inside the triangle for a total of 21 marked points.
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RESPONSE --> This was my answer. I looked at the totals from the other triangles and added the orig points if the process was repeated.
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19:51:03 `q005. The sequence of marked points is 3, 6, 10, 15, 21. What do expect will be the next number in this sequence?
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RESPONSE --> I expect it to be 27. This is because each one is going up in a sequence order.
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19:51:17 The differences between these numbers are 3, 4, 5, 6. The next difference, according to this pattern, should be 7, which would make the next number 28.
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RESPONSE --> This is the answer I got and the reasoning I was offering.
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19:52:16 `q006. How can you tell, in terms of the process you used to construct these triangles, that the next number should be 7 greater?
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RESPONSE --> Because each one has been going up a certain sequence. The one before it went up by six, the one before that went up by five, so if that pattern continued, the next would have gone up by seven.
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19:52:51 When you extend the triangle again, you will add two new endpoints and each side will now have 7 points. The 7 points on the new triangle will be all of the new points.
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RESPONSE --> This process of the drawing was throwing me, so I looked at the numbers instead to come up with my answers.
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19:53:38 `q007. How do you know this sequence will continue in this manner?
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RESPONSE --> That is the manner it had used throughout the sequence, so if we know that it is indeed a sequence, we can logically conclude that it would continue in this manner.
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19:54:05 Each time you extend the triangle, each side increases by 1. All the new marked points are on the new side, so the total number of marked points will increase by 1 more than with the previous extension.
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RESPONSE --> This is basically the way I was thinking, so no detailed critique is needed.
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?E}?????????????assignment #008 008. Arithmetic Sequences Liberal Arts Mathematics I 06-20-2007
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19:55:59 `q001. There are seven questions in this set. See if you can figure out a strategy for quickly adding the numbers 1 + 2 + 3 + ... + 100, and give your result if you are successful. Don't spend more than a few minutes on your attempt.
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RESPONSE --> The way I would do it would be to just add all the numbers in the sequence, but that would not qualify as ""quickly"" confidence assessment: 0
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19:57:25 These numbers can be paired as follows: 1 with 100, 2 with 99, 3 with 98, etc.. There are 100 number so there are clearly 50 pairs. Each pair adds up to the same thing, 101. So there are 50 pairs each adding up to 101. The resulting sum is therefore total = 50 * 101 = 5050.
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RESPONSE --> I would no have thought to pair them up to know that each one adds up to 101. If I had been able to do this, I feel certain that I would have been able to get to the multiplication step. self critique assessment: 2
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19:58:24 `q002. See if you can use a similar strategy to add up the numbers 1 + 2 + ... + 2000.
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RESPONSE --> 1000*2001 confidence assessment: 2
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19:58:54 Pairing 1 with 2000, 2 with 1999, 3 with 1998, etc., and noting that there are 2000 numbers we see that there are 1000 pairs each adding up to 2001. So the sum is 1000 * 2001 = 2,001,000.
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RESPONSE --> this is the process that I used, but I did not actually put the sum of the numbers. self critique assessment: 2
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20:01:03 `q003. See if you can devise a strategy to add up the numbers 1 + 2 + ... + 501.
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RESPONSE --> For this one I would do the same, but I came out with a dec in the problem. 501/2=250.5 250.5*50 confidence assessment: 0
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20:02:06 We can pair 1 with 501, 2 with 500, 3 with 499, etc., and each pair will have up to 502. However there are 501 numbers, so not all of the numbers can be paired. The number in the 'middle' will be left out. However it is easy enough to figure out what that number is, since it has to be halfway between 1 and 501. The number must be the average of 1 and 501, or (1 + 501) / 2 = 502 / 2 = 266. Since the other 500 numbers are all paired, we have 250 pairs each adding up to 502, plus 266 left over in the middle. The sum is 250 * 502 + 266 = 125,500 + 266 = 125,751. Note that the 266 is half of 502, so it's half of a pair, and that we could therefore say that we effectively have 250 pairs and 1/2 pair, or 250.5 pairs. 250.5 is half of 501, so we can still calculate the number of pairs by dividing the total number of number, 501, by 2. The total sum is then found by multiplying this number of pairs by the sum 502 of each pair: 250.5 * 502 = 125,766.
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RESPONSE --> This is much more clear. I did not take it far enough in my work to realize that not all the numbers could be paired successfully. confidence assessment: 0
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20:03:06 `q004. Use this strategy to add the numbers 1 + 2 + ... + 1533.
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RESPONSE --> 1533/2 766.5*1524 confidence assessment: 2
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20:04:13 Pairing the numbers, 1 with 1533, 2 with 1532, etc., we get pairs which each adult to 1534. There are 1533 numbers so there are 1533 / 2 = 766.5 pairs. We thus have a total of 1534 * 766.5, whatever that multiplies out to (you've got a calculator, and I've only got my unreliable head).
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RESPONSE --> These are the steps I've followed to get the answer. I may not have increased the whole number enough when working it out, I do not remember for sure. self critique assessment: 2
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20:05:27 `q005. Use a similar strategy to add the numbers 55 + 56 + ... + 945.
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RESPONSE --> 945/2=472.5 472.5*948=4479300 confidence assessment: 2
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20:06:58 We can pair up 55 and 945, 56 and 944, etc., obtaining 1000 for each pair. There are 945 - 55 + 1 = 891 numbers in the sum (we have to add 1 because 945 - 55 = 890 tells us how many 1-unit 'jumps' there are between 55 and 945--from 55 to 56, from 56 to 57, etc.. The first 'jump' ends up at 56 and the last 'jump' ends up at 945, so every number except 55 is the end of one of the 890 'jumps'. But 55 is included in the numbers to be summed, so we have 890 + 1 = 891 numbers in the sum). If we have 891 numbers in the sum, we have 891/2 = 445.5 pairs, each adding up to 1000. So we have a total of 445.5 * 1000 = 445,500.
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RESPONSE --> This one threw me a little apparently. If we have the same pattern as before, then I can pretty much get it, but this one is a little more tricky and I'm not entirely sure what I did wrong without looking back at my sequence. self critique assessment: 1
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20:07:56 `q006. Devise a strategy to add the numbers 4 + 8 + 12 + 16 + ... + 900.
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RESPONSE --> 900/4 = 225 225*904 confidence assessment: 2
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20:09:24 Pairing 4 with 900, 8 with 896, etc., we get pairs adding up to 904. The difference between 4 and 900 is 896. The numbers 'jump' by 4, so there are 896 / 4 = 224 'jumps'. None of these 'jumps' ends at the first number so there are 224 + 1 = 225 numbers. Thus we have 225 / 2 = 112.5 pairs each adding up to 904, and our total is 112.5 * 904.
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RESPONSE --> AHHH. On this one, I made the mistake of dividing my number by four instead of 2, otherwise my answer would have come out correctly. I also left out the subtraction step. self critique assessment: 2
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20:10:56 `q007. What expression would stand for the sum 1 + 2 + 3 + ... + n, where n is some whole number?
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RESPONSE --> I'm not sure how to write the expression because the set continues with no whole number in the given information confidence assessment: 0
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20:11:54 We can pair 1 and n, 2 and n-1, 3 and n-2, etc., in each case obtaining a sum of n + 1. There are n numbers so there are n/2 pairs, each totaling n + 1. Thus the total is n/2 * (n+1).
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RESPONSE --> I thought we needed to have a set ending number set to complete this. I have trouble with the abstract word problems. self critique assessment: 2
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