inclusionAI/Ring-lite-rl-data huggingface dataset:
Let $ABCD$ be a square with side length $1$. It is folded along a line $\ell$ that divides the square into two pieces with equal area. The minimum possible area of the resulting shape is $A$. Find the integer closest to $100A$.

Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50}
Suppose a tangent to the ellipse \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) intersects the ellipse \(\frac{x^2}{c^2} + \frac{y^2}{d^2} = 1\) at points \(P\) and \(Q\). If the tangents drawn to the second ellipse at these points are perpendicular, find the value of \(\frac{a^2}{c^2} + \frac{b^2}{d^2}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
For certain ordered pairs $(a,b)\,$ of real numbers, the system of equations
\[\begin{aligned} ax+by&=1 \\ x^2 + y^2 &= 50  \end{aligned}\]has at least one solution, and each solution is an ordered pair $(x,y)\,$ of integers. How many such ordered pairs $(a,b)\,$ are there?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{72}
Evaluate the real integral \( \int_{-\infty}^{\infty}{\frac{\sin x}{x^4-6x^2+10}\,\mathrm dx} \) using the Residue Theorem.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
In acute \( \bigtriangleup  {ABC} \) ,let \( I \) denote the incenter and suppose that line \( {AI} \) intersects segment \( {BC} \) at a point \( D \) . Given that
 \( {AI} = 3,{ID} = 2 \) ,and \( B{I}^{2} + C{I}^{2} = {64} \) ,compute \( B{C}^{2} \) .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
\[ {\dfrac{272}{3}} \]}
Two circles $F$ and $G$ with radius $10$ and $4$ respectively are externally tangent. A square $ABMC$ is inscribed in circle $F$ and equilateral triangle $MOP$ is inscribed in circle $G$ (they share vertex $M$). If the area of pentagon $ABOPC$ is equal to $a + b\sqrt{c}$, where $a$, $b$, $c$ are integers $c$ is square free, then find $a + b + c$.

Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{295}
A batch of one hundred parts is subjected to a sampling inspection. The entire batch is considered unacceptable if at least one defective part is found among the four parts that are tested. What is the probability that this batch will not be accepted if it contains $3\%$ defective parts?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.1164}
A [i]lame king[/i] is a chess piece that can move from a cell to any cell that shares at least one vertex with it, except for the cells in the same column as the current cell.

A lame king is placed in the top-left cell of a $7\times 7$ grid. Compute the maximum number of cells it can visit without visiting the same cell twice (including its starting cell).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{42}
Consider the endomorphism \( \phi: C(\mathbb{R}) \to C(\mathbb{R}) \) defined by \( \phi(f)(x) = x^2 f(x) \). Determine whether this endomorphism has an infinite norm, where the norm of a function \( f \) is defined as \( \|f\| = \sup_{x \in \mathbb{R}} |f(x)| \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Evaluate the limit: $$\lim _{x\to \infty }\frac{\left((x+2)!\right)^2\left(2\left(x-1\right)\right)!}{\left(\left(x+1\right)!\right)^2\left(2x\right)!}$$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{4}}
Given the equation \( \bar{L}\bar{C}=T^{-1}LCT \), where \( T \) is a coordinate change matrix, \( L \) is a column vector, and \( C \) is a row vector of compatible dimension, solve for the vector \( L \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{L = T \bar{L}}
Determine the fundamental group of a 2-sphere with the identification $x = -x$. What is the fundamental group of this space?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{Z}/2\mathbb{Z}}
Let $T$ denote the sum of all three-digit positive integers where each digit is different and none of the digits are 5. Calculate the remainder when $T$ is divided by $1000$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{840}
If the exponential function $y=(a^{2}-1)^{x}$ is a decreasing function on $(-\infty,+\infty)$, then the range of the real number $a$ is __________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-\sqrt{2}, -1) \cup (1, \sqrt{2})}
Determine a suitable substitution to simplify the integral \( \iint_R (x+y+1) \, dy \, dx \), where the region \( R \) is bounded by the curves \( y=1-x \), \( y=2-x \), \( y=1+x \), and \( y=-1+x \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{u = x + y, v = y - x}
Let $V,W \subset \mathbb{C}$ be open sets. Let $f\colon V \to W$ be a bijective holomorphic map. Determine if the set $Z:= \left\{v\in V \colon f'(v)=0\right\}$ is empty.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\emptyset}
Consider the following 100 sets of 10 elements each:
 \( \{ 1,2,3,\cdots ,{10}\} \) ,
 \( \{ {11},{12},{13},\cdots ,{20}\} \) ,
 \( \{ {21},{22},{23},\cdots ,{30}\} \) ,
 \( \vdots \)
 \( \{ {991},{992},{993},\cdots ,{1000}\} \) .
 How many of these sets contain exactly two multiples of 7 ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
42
}
In triangle $\triangle ABC$, given that $\cos A \cos B \cos C = \frac{1}{3}$, determine if the value of $\tan A \tan B + \tan B \tan C + \tan C \tan A$ is constant.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Determine whether the sequence \((a_n)_{n \in \mathbb{N}}\) defined by \(a_n = \sqrt[n]{|\sin(n)|}\) converges to 1.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Several irrational numbers are written on a blackboard. It is known that for every two numbers  $ a$  and  $ b$  on the blackboard, at least one of the numbers  $ a\over b\plus{}1$  and  $ b\over a\plus{}1$  is rational. What maximum number of irrational numbers can be on the blackboard? 

*Author: Alexander Golovanov*
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
计算下列实变函数定积分．

$\int _ { 0 } ^ { 2 \pi } \frac { d x } { ( 1 + \varepsilon \cos x ) ^ { 2 } } ( 0 < \varepsilon < 1 ) ,$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$2 \pi / ( 1 - \varepsilon ^ { 2 } ) ^ { 3 / 2 }$}
In the Cartesian coordinate plane \( xOy \), there are infinitely many circles inside the parabola \( y = ax^2 \) (where \( a \) is a constant). The centers of the circles \( O_1, O_2, \cdots \) are all on the y-axis. For each integer \( n > 1 \), the circle \( \odot O_n \) is tangent to the parabola and also externally tangent to the circle \( \odot O_{n-1} \). If the radius of \( \odot O_1 \) is \( \frac{1}{2a} \) and it passes through the vertex of the parabola, find the radius of \( \odot O_{2016} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{4031}{2 a}}
A school decides to check the homework of students whose student ID numbers end in 5 from each class. What type of sampling method is being applied? Provide your answer as the name of the sampling method.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{Systematic Sampling}
At a certain high school, the senior class is twice the size of the junior class. If 3/8 of the seniors study Japanese and 1/4 of the juniors study Japanese, while an additional 1/6 of the seniors and 1/12 of the juniors study both Japanese and French, what fraction of the students in both classes study Japanese, accounting for those who study both languages?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1/3}
In the right triangle \( \triangle ABC \), \( \angle B = 90^\circ \). Point \( P \) is on the angle bisector of \( \angle A \) within \( \triangle ABC \). Point \( M \) (distinct from \( A \) and \( B \)) is a point on side \( AB \). The lines \( AP \), \( CP \), and \( MP \) intersect sides \( BC \), \( AB \), and \( AC \) at points \( D \), \( E \), and \( N \) respectively. Given that \( \angle MPB = \angle PCN \) and \( \angle NPC = \angle MBP \), find the ratio \( \frac{S_{\triangle APC}}{S_{ACDE}} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
Someone draws a 2400 franc annuity 8 times at the end of every third year, what is the present value of the annuity at $4 \%$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11723}
Determine if there exists a largest nested prime number, and if so, find it.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{739397}
If $11 = x^6 + \frac{1}{x^6}$, find the value of $x^3 + \frac{1}{x^3}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{13}}
G3.3 $A B C D$ is a cyclic quadrilateral. $A C$ and $B D$ intersect at $G$. Suppose $A C=16 \mathrm{~cm}, B C=C D=8 \mathrm{~cm}$, $B G=x \mathrm{~cm}$ and $G D=y \mathrm{~cm}$. If $x$ and $y$ are integers and $x+y=c$, find the value of $c$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
Let \( l = \lim_{x \rightarrow a}\frac{x^x-a^x}{x-a} \) and \( m = \lim_{x \rightarrow a}\frac{a^x-x^a}{x-a} \) where \( a > 0 \). If \( l = m \), find the value of \( a \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e^2}
The parabola $y=-\frac{1}{3}x^{2}+2$, when $-1\leqslant x\leqslant 5$, the range of $y$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\frac{19}{3}\leqslant y\leqslant 2}
4. The base of the pyramid is a parallelogram. A plane is drawn through one side of the base and the midline of the opposite lateral face. Into what ratio does this plane divide the volume of the pyramid?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5:3}
Determine the cardinality of the set \( A = \{ [a, b] \mid a, b \in \mathbb{R}, \frac{a+b}{2} \in \mathbb{Q} \} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathfrak{c}}
Determine the most general form of the entire function $f(z)$ that satisfies the limit condition: \[ \lim_{z \to \infty} \frac{f(z)^2}{z^3} = 0. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{f(z) = az + b}
Determine the number of homomorphisms from the group $\Bbb Z/3\Bbb Z \times \Bbb Z/3\Bbb Z$ to the symmetric group $S_4$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{33}
There are two piles of stones, one containing 15 stones and the other containing 20 stones. Two players take turns in the following game: on each turn, a player can take any number of stones, but only from one pile. The player who cannot make a move loses. Who wins with optimal play?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{First Player}
How many integer triples $(x,y,z)$ are there satisfying $x^3+y^3=x^2yz+xy^2z+2$ ?

Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
4}
Call a polynomial $f$ with positive integer coefficients "triangle-compatible" if any three coefficients of $f$ satisfy the triangle inequality. For instance, $3x^3 + 4x^2 + 6x + 5$ is triangle-compatible, but $3x^3 + 3x^2 + 6x + 5$ is not. Given that $f$ is a degree $20$ triangle-compatible polynomial with $-20$ as a root, what is the least possible value of $f(1)$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{420}
Evaluate the limit: $$\lim_{n \rightarrow \infty} \sqrt{n}\int_{-1/\sqrt{n}}^0  e^{x^2/2} \text{erfi}\left(\frac{1}{\sqrt{2n}}\right) dx$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
How many elements in the set \( S = \{0, 1, \ldots, 5399\} \) are quadratic residues modulo 5400?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{363}
Given \( a = 19911991 \cdots \cdots 1991 \) (a repeated sequence of 1991), what is the remainder when \( a \) is divided by 13?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Last week, a farmer shipped 15 boxes of pomelos which had 360 pomelos in total. The boxes were of three types: small boxes containing 10 pomelos each, medium boxes containing 20 pomelos each, and large boxes containing 30 pomelos each. This week, the farmer shipped 25 boxes: 10 of them were small boxes, 8 were medium boxes, and 7 were large boxes. During transportation, 3 small boxes, 2 medium boxes, and 2 large boxes were damaged, causing a loss of 10% of pomelos for small boxes, 15% for medium boxes, and 20% for large boxes. How many dozens of pomelos did the farmer successfully ship in total for both weeks?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{67}
A position of the hands of a (12-hour, analog) clock is called valid if it occurs in the course of a day. For example, the position with both hands on the 12 is valid; the position with both hands on the 6 is not. A position of the hands is called bivalid if it is valid and, in addition, the position formed by interchanging the hour and minute hands is valid. Find the number of bivalid positions.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{143}
Wendy is playing darts with a circular dartboard of radius 20. Whenever she throws a dart, it lands uniformly at random on the dartboard. At the start of her game, there are 2020 darts placed randomly on the board. Every turn, she takes the dart farthest from the center, and throws it at the board again. What is the expected number of darts she has to throw before all the darts are within 10 units of the center?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6060}
Two concentric circles have radii of 1 and 2 units, respectively. Determine the minimum possible area of a cyclic quadrilateral that can be inscribed in the larger circle while also containing the smaller circle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3\sqrt{3}}
Find \( F''(\pi) \) if \[ F(x) = \int_x^{\sin x}  \left(\int_0^{\sin t} (1+u^4)^{0.5} \,du\right)\,dt \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
We have a half-circle with endpoints $A$ and $B$ and center $S$. The points $C$ and $D$ lie on the half-circle such that $\angle BAC = 20^\circ$ and the lines $AC$ and $SD$ are perpendicular to each other. If the angle between the lines $AC$ and $BD$ is $x^\circ$, what is the value of $x$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{55}
Three cones with bases touching each other are placed on a table. The heights of the cones are equal, and the radii of their bases are 1, 2, and 3. A sphere is placed on the table such that it touches all three cones. It turns out that the center of the sphere is equidistant from all points of contact with the cones. Find the radius of the sphere.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
A right pyramid has a rectangular base with dimensions 8 cm by 15 cm. Its peak is 10 cm vertically above the center of its base. Calculate the sum of the lengths of the pyramid's eight edges. Round your answer to the nearest whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{99}
Find the sum of the series $$\sum_{n=1}^{\infty}\frac{(-n)^{n-1}e^{-n}}{n!}$$ using the Lambert W function.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{W\left(\frac{1}{e}\right)}
The Planar National Park is an undirected 3-regular planar graph (i.e., all vertices have degree 3). A visitor walks through the park as follows: she begins at a vertex and starts walking along an edge. When she reaches the other endpoint, she turns left. On the next vertex, she turns right, and so on, alternating left and right turns at each vertex. She does this until she gets back to the vertex where she started. What is the largest possible number of times she could have entered any vertex during her walk, over all possible layouts of the park?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Determine the side length of the square base of a square pyramid with equilateral triangular faces that can completely contain a cuboid with dimensions 25 (height) x 50 (width) x 50 (length), given that the bottom of the cuboid is 26 units above the base of the pyramid.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50 + 51\sqrt{2}}
Let \( F_1 \) and \( F_2 \) be the left and right foci of the ellipse \( C \):

\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \quad (a > b > 0) \]

If there exists a point \( P \) on the ellipse \( C \) such that \( P F_1 \) is perpendicular to \( P F_2 \), then determine the range of the eccentricity \( e \) of the ellipse.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left[ \frac{\sqrt{2}}{2}, 1 \right)}
Given that $U$ and $V$ are independent standard normal variables, calculate $\text{Cov}(U^2 - V^2 \mid U > 0)$. Provide your answer as a single value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Transform a rectangle with a length of 10 cm and a width of 8 cm into a parallelogram with a height of 9 cm. The area of this parallelogram is \_\_\_\_\_\_ square centimeters.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{72}
In heptagon $GEOMETRY$, $\angle G \cong \angle E \cong \angle T \cong \angle R$ and $\angle M$ is supplementary to $\angle Y$, $\angle J$ is supplementary to $\angle O$. Find the measure of $\angle T$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{135^\circ}
Find the maximum number of colors that can be used to paint the edges of a cube (each edge in one color) such that for every pair of colors, there exist two adjacent edges painted in these colors. Edges are considered adjacent if they share a common vertex.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Given a language with $n+1$ atomic propositions, determine the number of non-equivalent propositions that can be expressed using a combinatorial approach.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{2^{n+1}}}
Find the minimum value of the function \( f(x) = \sum_{i=1}^{2017} i |x - i| \) for \( x \in [1, 2017] \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{801730806}
Given the sum \( \sum_{j=1}^M c_M(j) \), where \( \lim_{M\to \infty}c_M(j)=0 \) and there exists a sequence \( d_j \ge 0 \) with \( \sum_{j=1}^\infty d_j < \infty \) such that \( |c_M(j)| \le d_j \) for all \( j \) and \( M \), determine if the sum converges to zero as \( M\to \infty \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Evaluate the double integral \( \int_0^\pi \int_0^\pi |\cos(x+y)| \,dx\,dy \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\pi}
Evaluate the triple integral \( \iiint_R z(x^2 + y^2 + z^2)^{-3/2} \, dV \) over the region \( R \) within the sphere \( x^2 + y^2 + z^2 \leq 81 \) and where \( z \geq 4.5 \), using spherical coordinates.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{9\pi}{4}}
Calculate the volume of the body bounded by the surface defined by the inequality:
\[ 0 \leq z \leq c \cdot \sin\left(\pi \sqrt{\frac{x^2}{a^2} + \frac{y^2}{b^2}}\right) \]
where \(a\), \(b\), and \(c\) are positive parameters.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2abc}
Evaluate the integral \( \int_0^\infty \frac{\sin^2(4\pi x)}{x^{3/2}}dx. \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\pi}
A digital display shows the current date as an $8$-digit integer consisting of a $4$-digit year, followed by a $2$-digit month, followed by a $2$-digit date within the month. For example, Arbor Day this year is displayed as 20230428. For how many dates in $2023$ will each digit appear an even number of times in the 8-digital display for that date?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$9$}
Calculate the flow of the vector field \( F(x,y,z) = (xz, yz, -z^2) \) out of the surface \( S = \{(x,y,z) \in \mathbb{R}^3 : z^2 = 1 + x^2 + y^2, 2 \leq z \leq 3\} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60\pi}
Determine the expected size of the largest connected component of the graph $H(s_1, s_2)$ as a function of $n$, where $s_1$ and $s_2$ are two binary strings of length $n$. The graph $H(s_1, s_2)$ is constructed based on the principles of Hitomezashi Sashiko, a form of decorative mending.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\Theta(n)}
Solve for \( x \) in the equation: \[ \frac{(1+x)^{36} -1}{x} = \frac{20142.9}{420} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.0158}
Let \( f: \mathbb{C}[x,y] \rightarrow \mathbb{C}[t] \) be a non-zero homomorphism such that \( \ker f \) strictly contains a prime ideal \( P \neq 0 \). Determine whether \( \text{im} \ f = \mathbb{C} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{C}}
Calculate the variance of the random variable $X$ given that $Y=25$ for the joint probability density function $f(x,y) = \frac{1}{1250}$, where $0 < x < y < 50$. Provide your answer as a single numerical value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{625}{12}}
Determine the Lebesgue measure of the set \( \left\{ \frac{x}{y} : x, y \in I_\alpha^* \right\} \), where \( I_\alpha \subset [0,1] \) is an \( \alpha \)-Cantor set of Lebesgue measure \( \alpha \) and \( I_\alpha^* = I_\alpha \setminus \{0\} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
In the context of Fourier series, can the coefficients of the full Fourier series always be obtained from the complex form coefficients \( C_n \)? Answer "yes" or "no."
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{yes}
How many sequences of words (not necessarily grammatically correct) have the property that the first word has one letter, each word can be obtained by inserting a letter somewhere in the previous word, and the final word is CCAMT? Here are examples of possible sequences:
\[C,CA,CAM,CCAM,CCAMT.\]
\[A,AT,CAT,CAMT,CCAMT.\]

Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60}
Consider the function \( f(y) = \int_0^\epsilon \frac{\left(\frac{p-t}{p(1-t)}\right)^y-(1-t)^{y(1-p)/p}}{t\log(1-t)}dt \) for \( p \in (0,1) \) and \( \epsilon > 0 \) a small constant. Is it true that \( f(y) = O(1) \) for large \( y \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{O(1)}
Example 5 If for any positive real numbers, $\frac{a^{2}}{\sqrt{a^{4}+3 b^{4}+3 c^{4}}}+\frac{k}{a^{3}} \cdot\left(\frac{c^{4}}{b}+\frac{b^{4}}{c}\right) \geqslant \frac{2 \sqrt{2}}{3}$. Always holds, find the minimum value of the real number $k$.


Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{\sqrt[4]{24}}}