Datasets:

1231czx
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9B_iter4_bmpmodel_onpolicy

gt stringlengths 1 88	chosen listlengths 4 12	rejected listlengths 4 12	rej sequencelengths 1 1
32	[ { "content": "If the first popsicle in a box of six popsicles melts at a certain rate, and each subsequent popsicle melts twice as fast as the previous one, how many times faster does the last remaining popsicle melt compared to the first one?", "role": "user" }, { "content": "Let's say the melting ...	[ { "content": "If the first popsicle in a box of six popsicles melts at a certain rate, and each subsequent popsicle melts twice as fast as the previous one, how many times faster does the last remaining popsicle melt compared to the first one?", "role": "user" }, { "content": "Let's define the speed...	[ "16" ]
9	[ { "content": "If $z$ and $w$ are complex numbers such that $\|2z - w\| = 25$, $\|z + 2w\| = 5$, and $\|z + w\| = 2$, what is the value of $\|z\|$?", "role": "user" }, { "content": "Let's write down the equations based on the given information and solve for $\|z\|$.\n```python\nfrom sympy import symbols, Eq, s...	[ { "content": "If $z$ and $w$ are complex numbers such that $\|2z - w\| = 25$, $\|z + 2w\| = 5$, and $\|z + w\| = 2$, what is the value of $\|z\|$?", "role": "user" }, { "content": "Let's use sympy to solve for $z$.\n```python\nfrom sympy import symbols, sqrt, solve\n\n# complex numbers\nz, w = symbols('z w'...	[ "\\sqrt{130}" ]
10	[ { "content": "Calculate the total prime divisors of the number obtained by multiplying every integer from 1 to 30, inclusive.", "role": "user" }, { "content": "Let's first define a function to calculate the prime divisors of a number.\n```python\nimport math\n\ndef find_prime_divisors(n):\n divis...	[ { "content": "Calculate the total prime divisors of the number obtained by multiplying every integer from 1 to 30, inclusive.", "role": "user" }, { "content": "First, let's calculate the number 1 * 2 * 3 * ... * 30\n```python\n# Calculate the product of integers from 1 to 30\nimport math\nproduct = ...	[ "2332800" ]
51	[ { "content": "What is the remainder when $7^{2012} + 7^{2010} + 7^{2008}$ is divided by $100$?", "role": "user" }, { "content": "Let's write a function in sympy that computes the remainder of a number modulo 100.\n```python\nfrom sympy import Mod\n\n# write a function that computes the remainder of ...	[ { "content": "What is the remainder when $7^{2012} + 7^{2010} + 7^{2008}$ is divided by $100$?", "role": "user" }, { "content": "To answer this, we can use Python's mod operator (%) to perform modular arithmetic, and look for patterns in the remainders when increasing powers of 7 are divided by 100....	[ "3" ]
5	[ { "content": "What is the remainder when $5^{2021}$ is divided by 6?", "role": "user" }, { "content": "Let's write a modular arithmetic function for sympy.\n```python\nfrom sympy import mod_inverse\n\ndef my_mod(a, b):\n if a == 0:\n return 0\n elif a > 0:\n return a % b\n els...	[ { "content": "What is the remainder when $5^{2021}$ is divided by 6?", "role": "user" }, { "content": "To solve this problem, we can use Python and the sympy library.\nWe can use the factorial function to calculate the remainder when dividing $5^{2021}$ by 6.\n```python\nfrom sympy import factorial\...	[ "1" ]
4	[ { "content": "An arithmetic sequence has a common difference of 2, and its 50th term is 100. What is the second term?", "role": "user" }, { "content": "We can use the following formula to find the $n$th term of an arithmetic sequence with common difference $d$ and first term $a$,\n\\[a_n = a + (n-1)...	[ { "content": "An arithmetic sequence has a common difference of 2, and its 50th term is 100. What is the second term?", "role": "user" }, { "content": "Let's define the sequence using sympy and extract the second term of the sequence.\n```python\nimport sympy as sp\n\nx = sp.symbols('x')\n\nseq = sp...	[ "6" ]
4	[ { "content": "In base-7, how many places are needed to represent the decimal number nine hundred and fifty-six?", "role": "user" }, { "content": "To solve this problem, we first convert 956 to base 7, and then count the number of digits.\n```python\ndef convert_to_base(num, base):\n if num == 0:\...	[ { "content": "In base-7, how many places are needed to represent the decimal number nine hundred and fifty-six?", "role": "user" }, { "content": "I will find the answer by calculating the number of digits in the base-7 expansion of 956 and adding one.\n```python\nfrom sympy import floor\n\nbase = 7\...	[ "5" ]
7680	[ { "content": "A chess club hosts a single-elimination tournament with 256 members, and each member pays an entry fee of $10. Additionally, the club charges a membership fee of $20 per member annually. Calculate the total amount of money collected from entry fees and membership fees.", "role": "user" }, ...	[ { "content": "A chess club hosts a single-elimination tournament with 256 members, and each member pays an entry fee of $10. Additionally, the club charges a membership fee of $20 per member annually. Calculate the total amount of money collected from entry fees and membership fees.", "role": "user" }, ...	[ "7670" ]
15	[ { "content": "If a bear consumes up to 90 pounds of food per day in preparation for hibernation, and Victor weighs 126 pounds, how many times Victor's weight worth of food would a bear eat in a span of 3 weeks?", "role": "user" }, { "content": "Let's solve this problem using Python code.\n```python\...	[ { "content": "If a bear consumes up to 90 pounds of food per day in preparation for hibernation, and Victor weighs 126 pounds, how many times Victor's weight worth of food would a bear eat in a span of 3 weeks?", "role": "user" }, { "content": "Let's solve this problem using Python code.\n```python\...	[ "1890" ]
29	[ { "content": "If p is a prime number such that p = x^2 + 4 for some integer x, and p = y^2 - 7 for some integer y, what is the value of p?", "role": "user" }, { "content": "Let's write down all possible pairs $(x,y)$ and filter out those that satisfy $p = x^2 + 4$ and $p = y^2 - 7$.\nThen we can loo...	[ { "content": "If p is a prime number such that p = x^2 + 4 for some integer x, and p = y^2 - 7 for some integer y, what is the value of p?", "role": "user" }, { "content": "Let's list out all primes up to 100 and for each one try to find $x$ and $y$ such that $p = x^2 + 4$ and $p = y^2 - 7$.\n\nWe c...	[ "131" ]
35	[ { "content": "Identify the largest factor that is common to both 2835 and 8960.", "role": "user" }, { "content": "Let's use sympy to get the GCD of these two numbers.\n```python\nimport sympy as sp\n\n# Define the two numbers\nnum1 = 2835\nnum2 = 8960\n\n# find the greatest common divisor of num1 an...	[ { "content": "Identify the largest factor that is common to both 2835 and 8960.", "role": "user" }, { "content": "Let's list out the factors of 2835 and 8960.\nFor each pair of factors, we check if they are common to both 2835 and 8960.\n```python\nfactors_2835 = set(range(1, 2835 + 1))\nfactors_896...	[ "5" ]
3	[ { "content": "What is the number $y$ that meets the condition given by the equation $\|y-6\| + 2y = 9$?", "role": "user" }, { "content": "Let's use sympy to solve for $y$.\n```python\nfrom sympy import symbols, Abs, solve\n\ny = symbols('y')\n\nconditions = Abs(y - 6) + 2 * y - 9\nsolve(conditions)\n`...	[ { "content": "What is the number $y$ that meets the condition given by the equation $\|y-6\| + 2y = 9$?", "role": "user" }, { "content": "Let's solve this equation using sympy.\n```python\nimport sympy as sp\n\n# let's define the variable\ny = sp.symbols('y')\n\n# solve the equation\neq = sp.Eq(sp.Abs...	[ "5" ]
360	[ { "content": "Alex has 10 different kinds of lunch meat and 9 different kinds of cheese. If he wants to make a sandwich with one kind of meat and two kinds of cheese, how many different sandwiches could he make? (It does not matter in which order he chooses the two types of cheese.)", "role": "user" }, ...	[ { "content": "Alex has 10 different kinds of lunch meat and 9 different kinds of cheese. If he wants to make a sandwich with one kind of meat and two kinds of cheese, how many different sandwiches could he make? (It does not matter in which order he chooses the two types of cheese.)", "role": "user" }, ...	[ "90" ]
0	[ { "content": "Calculate the slope of the line that passes through the midpoints of two line segments: the first with endpoints (0, 0) and (2, 3), and the second with endpoints (5, 0) and (6, 3). Simplify your answer.", "role": "user" }, { "content": "Let's find the equation of the line that passes t...	[ { "content": "Calculate the slope of the line that passes through the midpoints of two line segments: the first with endpoints (0, 0) and (2, 3), and the second with endpoints (5, 0) and (6, 3). Simplify your answer.", "role": "user" }, { "content": "Let's calculate the midpoints $P$ and $Q$ of the ...	[ "\\frac{1}{3}" ]
9	[ { "content": "If twice the amount of money Jericho has is 60 and he owes $14 to Annika and half as much to Manny, how much money will Jericho have left after paying off all his debts?", "role": "user" }, { "content": "Let's solve this problem using Python code.\n```python\n# let's denote the amount ...	[ { "content": "If twice the amount of money Jericho has is 60 and he owes $14 to Annika and half as much to Manny, how much money will Jericho have left after paying off all his debts?", "role": "user" }, { "content": "Let's solve this problem using Python code.\n```python\ntwice_jericho_money = 60\n...	[ "39" ]
-\frac{25}{4}	[ { "content": "In the standard form of a parabola, $a(x - h)^2 + k$, what is the constant term $k$ after transforming $x^2 - 5x$?", "role": "user" }, { "content": "We can use sympy library to complete the square of $x^2 - 5x$ and find the constant term in the resulting expression.\n```python\nfrom sy...	[ { "content": "In the standard form of a parabola, $a(x - h)^2 + k$, what is the constant term $k$ after transforming $x^2 - 5x$?", "role": "user" }, { "content": "Let's write down an equation for the parabola and solve for $k$ using sympy.\n```python\nimport sympy as sp\n\n# define the symbols\nx, h...	[ "0" ]
2018	[ { "content": "Let $x$ and $y$ be positive real numbers. Find the minimum value of\n\\[\\left( x + \\frac{1}{y} \\right) \\left( x + \\frac{1}{y} - 2018 \\right) + \\left( y + \\frac{1}{x} \\right) \\left( y + \\frac{1}{x} - X \\right).\\]\nIf we know the answer to the above question is -2036162, what is the va...	[ { "content": "Let $x$ and $y$ be positive real numbers. Find the minimum value of\n\\[\\left( x + \\frac{1}{y} \\right) \\left( x + \\frac{1}{y} - 2018 \\right) + \\left( y + \\frac{1}{x} \\right) \\left( y + \\frac{1}{x} - X \\right).\\]\nIf we know the answer to the above question is -2036162, what is the va...	[ "2035.01486620416" ]
60	[ { "content": "Tom broke his leg and needed to go to the doctor. The visit cost $300 and the cast cost $200. If insurance covered x% how much was Tom's out-of-pocket cost?\nIf we know the answer to the above question is 200, what is the value of unknown variable x?", "role": "user" }, { "content": ...	[ { "content": "Tom broke his leg and needed to go to the doctor. The visit cost $300 and the cast cost $200. If insurance covered x% how much was Tom's out-of-pocket cost?\nIf we know the answer to the above question is 200, what is the value of unknown variable x?", "role": "user" }, { "content": ...	[ "50" ]
\frac{640}{3}\pi	[ { "content": "Kadin has built a snowman using three spherical snowballs with radii measuring 2 inches, 3 inches, and 5 inches, respectively. How many cubic inches of snow are contained within the snowman if one assumes each snowball maintains a perfect spherical shape? The volume should be expressed as a functi...	[ { "content": "Kadin has built a snowman using three spherical snowballs with radii measuring 2 inches, 3 inches, and 5 inches, respectively. How many cubic inches of snow are contained within the snowman if one assumes each snowball maintains a perfect spherical shape? The volume should be expressed as a functi...	[ "213.333333333333\\pi" ]
7	[ { "content": "Determine the remainder when the consecutive odd numbers $11065, 11067, 11069, 11071, 11073, 11075$, and $11077$ are added together and the sum is divided by $14$.", "role": "user" }, { "content": "I will find the answer by calculating the sum and then dividing it by 14 and then findin...	[ { "content": "Determine the remainder when the consecutive odd numbers $11065, 11067, 11069, 11071, 11073, 11075$, and $11077$ are added together and the sum is divided by $14$.", "role": "user" }, { "content": "When we are trying to find the remainder of a sum, we can use the fact that for any inte...	[ "0" ]
2	[ { "content": "Blake bought 4 lollipops and 6 packs of chocolate. If each lollipop costs $x and a pack of chocolate costs the same as four lollipops, Blake will get back 4 change if he gave the cashier 6 $10 bills. What is the value of unknown variable x?", "role": "user" }, { "content": "Let's solv...	[ { "content": "Blake bought 4 lollipops and 6 packs of chocolate. If each lollipop costs $x and a pack of chocolate costs the same as four lollipops, Blake will get back 4 change if he gave the cashier 6 $10 bills. What is the value of unknown variable x?", "role": "user" }, { "content": "Let's solv...	[ "\\frac{1}{2}" ]
3	[ { "content": "Palindromes are numbers that read the same backwards and forwards, like 5665. The least possible positive four-digit palindrome that is divisible by X is 1221. What is the value of unknown variable X?", "role": "user" }, { "content": "Let's check all the possible values for X using Pyt...	[ { "content": "Palindromes are numbers that read the same backwards and forwards, like 5665. The least possible positive four-digit palindrome that is divisible by X is 1221. What is the value of unknown variable X?", "role": "user" }, { "content": "Let's solve this problem using Python's sympy libra...	[ "11" ]
67	[ { "content": "If Matt has six cats and half of them are female, and each female cat has 7 kittens, with 9 of them being sold, what is the approximate percentage of Matt's remaining cats that are kittens (rounded to the nearest percent)?", "role": "user" }, { "content": "Let's solve this problem usin...	[ { "content": "If Matt has six cats and half of them are female, and each female cat has 7 kittens, with 9 of them being sold, what is the approximate percentage of Matt's remaining cats that are kittens (rounded to the nearest percent)?", "role": "user" }, { "content": "Let's solve this problem usin...	[ "140" ]
160	[ { "content": "If Brianne saves $10 in January and each month she saves twice as much as her previous month's savings, how much will she save in May?", "role": "user" }, { "content": "Let's solve this problem using Python code.\n```python\njanuary_saving = 10\nsaving_per_month = 2\nnumber_of_months =...	[ { "content": "If Brianne saves $10 in January and each month she saves twice as much as her previous month's savings, how much will she save in May?", "role": "user" }, { "content": "Let's solve this problem using Python code.\n```python\nbase_saving = 10\ntotal_saved_in_may = 0\nfor i in range(5):\...	[ "310" ]
10	[ { "content": "Jerry finds six hawk feathers and 17 times as many eagle feathers as hawk feathers. If he gives x feathers to his sister and sells half the remaining feathers, He has 49 left. What is the value of unknown variable x?", "role": "user" }, { "content": "Let's solve this problem using Pyth...	[ { "content": "Jerry finds six hawk feathers and 17 times as many eagle feathers as hawk feathers. If he gives x feathers to his sister and sells half the remaining feathers, He has 49 left. What is the value of unknown variable x?", "role": "user" }, { "content": "Let's solve this problem using Pyth...	[ "20" ]
40	[ { "content": "A field in the shape of a trapezoid has a height of 80 feet, with one base being 60 feet and the other base twice as long. A line is drawn parallel to the bases, creating a smaller, similar trapezoid that has a height that is half the height of the original trapezoid. What is the height of this sm...	[ { "content": "A field in the shape of a trapezoid has a height of 80 feet, with one base being 60 feet and the other base twice as long. A line is drawn parallel to the bases, creating a smaller, similar trapezoid that has a height that is half the height of the original trapezoid. What is the height of this sm...	[ "20" ]
2	[ { "content": "Find all real numbers $k$ such that\n\\[\\left\\\| k \\begin{pmatrix} X \\\\ -3 \\end{pmatrix} - \\begin{pmatrix} 4 \\\\ 7 \\end{pmatrix} \\right\\\| = 2 \\sqrt{13}.\\]Enter all the solutions, separated by commas.\nIf we know the answer to the above question is -1, what is the value of unknown varia...	[ { "content": "Find all real numbers $k$ such that\n\\[\\left\\\| k \\begin{pmatrix} X \\\\ -3 \\end{pmatrix} - \\begin{pmatrix} 4 \\\\ 7 \\end{pmatrix} \\right\\\| = 2 \\sqrt{13}.\\]Enter all the solutions, separated by commas.\nIf we know the answer to the above question is -1, what is the value of unknown varia...	[ "-1" ]
64	[ { "content": "There was a two-stage sale on a television set. Initially, the price was cut by $40\\%$, and in the final clearance, the new price was reduced further by $40\\%$. Calculate the overall percentage reduction from the original price.", "role": "user" }, { "content": "Let's define `percent...	[ { "content": "There was a two-stage sale on a television set. Initially, the price was cut by $40\\%$, and in the final clearance, the new price was reduced further by $40\\%$. Calculate the overall percentage reduction from the original price.", "role": "user" }, { "content": "Let's consider the or...	[ "80" ]
46	[ { "content": "If Andy needs to raise the temperature of the candy mixture from 60 degrees to 240 degrees and then cool it down to 170 degrees, and the candy heats at a rate of 5 degrees per minute and cools at a rate of 7 degrees per minute, how many minutes will it take for the candy to be done?", "role": ...	[ { "content": "If Andy needs to raise the temperature of the candy mixture from 60 degrees to 240 degrees and then cool it down to 170 degrees, and the candy heats at a rate of 5 degrees per minute and cools at a rate of 7 degrees per minute, how many minutes will it take for the candy to be done?", "role": ...	[ "20" ]
200	[ { "content": "A hot air balloon with x balloons is blowing up. After about half an hour, 1/5 of the total number of balloons in the hot air balloon have blown up. After another hour, twice the number of balloons that had already blown up also blow up. 80 balloons in the hot air balloon remain intact. What is th...	[ { "content": "A hot air balloon with x balloons is blowing up. After about half an hour, 1/5 of the total number of balloons in the hot air balloon have blown up. After another hour, twice the number of balloons that had already blown up also blow up. 80 balloons in the hot air balloon remain intact. What is th...	[ "125" ]
20	[ { "content": "John hits 70% of his free throws. For every foul he gets 2 shots. He gets fouled 5 times a game. How many free throws does he get if he plays in 80% of the x games the team plays?\nIf we know the answer to the above question is 112, what is the value of unknown variable x?", "role": "user" ...	[ { "content": "John hits 70% of his free throws. For every foul he gets 2 shots. He gets fouled 5 times a game. How many free throws does he get if he plays in 80% of the x games the team plays?\nIf we know the answer to the above question is 112, what is the value of unknown variable x?", "role": "user" ...	[ "50" ]
\frac{1}{2}	[ { "content": "In Pascal's triangle, the 1st, 2nd, and 3rd rows are given as shown:\n\n\\[\n\\begin{array}{ccccccc}\n& & 1 & & 1 & & \\\\\n& 1 & & 2 & & 1 & \\\\\n1 & & 3 & & 3 & & 1\n\\end{array}\n\\]\n\nLet $(a_i),$ $(b_i),$ $(c_i)$ represent the sequence of elements in the 2005th, 2006th, and 2007th rows, res...	[ { "content": "In Pascal's triangle, the 1st, 2nd, and 3rd rows are given as shown:\n\n\\[\n\\begin{array}{ccccccc}\n& & 1 & & 1 & & \\\\\n& 1 & & 2 & & 1 & \\\\\n1 & & 3 & & 3 & & 1\n\\end{array}\n\\]\n\nLet $(a_i),$ $(b_i),$ $(c_i)$ represent the sequence of elements in the 2005th, 2006th, and 2007th rows, res...	[ "1" ]
3	[ { "content": "At a garage sale, Tish bought 4 items: a red horseshoe magnet, two stuffed animals, and a large sticker which read, \"Why can't teachers solve their own math problems?\" The magnet cost x times more than the sticker, but only one quarter the price of the two stuffed animals combined. If the Magn...	[ { "content": "At a garage sale, Tish bought 4 items: a red horseshoe magnet, two stuffed animals, and a large sticker which read, \"Why can't teachers solve their own math problems?\" The magnet cost x times more than the sticker, but only one quarter the price of the two stuffed animals combined. If the Magn...	[ "4" ]
729	[ { "content": "In a math competition, one of the questions is to find the value of the cube of the difference of the squares of the numbers 5 and 4. What should the contestants answer?", "role": "user" }, { "content": "To solve this problem, we first need to calculate the value of $5^2 - 4^2$.\n```py...	[ { "content": "In a math competition, one of the questions is to find the value of the cube of the difference of the squares of the numbers 5 and 4. What should the contestants answer?", "role": "user" }, { "content": "Let's use sympy to get the answer.\n```python\nimport sympy as sp\n\n# define the ...	[ "1" ]
\frac{1}{10}	[ { "content": "Solve for $y$ given that $x^3y - 2x^2y + xy = 101000$ if $x$ is equal to 101.", "role": "user" }, { "content": "Let's solve it using python:\n```python\nfrom sympy import symbols, Eq, solve\n\nx, y = symbols('x y')\n\n# Solve the equation\nsolution = solve(Eq(x*3y - 2x2y + x*y, 1...	[ { "content": "Solve for $y$ given that $x^3y - 2x^2y + xy = 101000$ if $x$ is equal to 101.", "role": "user" }, { "content": "Let's use sympy to solve this equation.\n```python\nfrom sympy import symbols, solve\n\n# define the variables\nx, y = symbols('x y')\n\n# equation\nequation = x*3 y - 2 *...	[ "100" ]
54	[ { "content": "Assuming there are positive integers $m$ and $n$, which together satisfy $m^2 + n$ being under 22, how many such integral pairs are possible?", "role": "user" }, { "content": "Let's enumerate all possible pairs of positive integers $m$ and $n$ such that $m^2 + n < 22$.\n```python\n# in...	[ { "content": "Assuming there are positive integers $m$ and $n$, which together satisfy $m^2 + n$ being under 22, how many such integral pairs are possible?", "role": "user" }, { "content": "Let's write a function `count_pairs` that takes as input an integer $n$ and counts the number of pairs $(m,n)$...	[ "4" ]
12\sqrt{2}	[ { "content": "What is the simplest form of the square root of 288?", "role": "user" }, { "content": "Let's use sympy to calculate the simplest form of the square root of 288.\n```python\nfrom sympy import sqrt\n\n# calculate square root of 288\nsqrt(288).factor()\n``````", "role": "assistant" ...	[ { "content": "What is the simplest form of the square root of 288?", "role": "user" }, { "content": "Let's break down the problem into smaller steps.\nFirst, we need to find the prime factors of 288.\nSecond, we need to group the prime factors to find the square root.\nThird, we need to simplify the...	[ "12" ]
12	[ { "content": "If you multiply the trinomial \\[(m+n+p)\\] with the quadrinomial \\[(q+r+s+t)\\], how many distinct terms will result from the expansion?", "role": "user" }, { "content": "We can compute this using Python.\n```python\nimport itertools\n\n# Because the question does not mention anyth...	[ { "content": "If you multiply the trinomial \\[(m+n+p)\\] with the quadrinomial \\[(q+r+s+t)\\], how many distinct terms will result from the expansion?", "role": "user" }, { "content": "Let's use sympy to calculate the number of terms in the expansion by calculating the number of elements in the ...	[ "7" ]
3	[ { "content": "Calculate the slope of the line tangent to the circle at the point (1,2) where the center of the circle is at (-2,3). Present your answer as a reduced fraction.", "role": "user" }, { "content": "Let's first check whether point (1,2) lies on the circle given that the center of the circl...	[ { "content": "Calculate the slope of the line tangent to the circle at the point (1,2) where the center of the circle is at (-2,3). Present your answer as a reduced fraction.", "role": "user" }, { "content": "Let's define the circle using sympy.\n```python\nfrom sympy import Point, Circle\nfrom symp...	[ "-4" ]
2051	[ { "content": "Take six numbers: 1867, 1993, 2019, 2025, 2109, and 2121. If the average of four of these numbers is 2008, calculate the average of the two numbers not included in finding the first average.", "role": "user" }, { "content": "Let's name these numbers A, B, C, D, E, and F.\nLet's now wri...	[ { "content": "Take six numbers: 1867, 1993, 2019, 2025, 2109, and 2121. If the average of four of these numbers is 2008, calculate the average of the two numbers not included in finding the first average.", "role": "user" }, { "content": "Let's store the given numbers in a list.\n```python\ngiven_nu...	[ "2070" ]