inclusionAI/Ring-lite-rl-data huggingface dataset:
There is a set of points \( M \) on a plane and seven different circles \( C_{1}, C_{2}, \cdots, C_{7} \). Circle \( C_{7} \) passes through exactly 7 points in \( M \), circle \( C_{6} \) passes through exactly 6 points in \( M \), and so on, with circle \( C_{1} \) passing through exactly 1 point in \( M \). Determine the minimum number of points in set \( M \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
设 3 阶实对称矩阵 A 的三个特征值分别为 $-1,-1,2$, 且 $(1,1,-1)^{T}$ 是特征值 2所对应的特征向量。记 A中所有元素的平方之和为 I, 则 $[I]=$ $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Five students named Fred, Gail, Henry, Iggy, and Joan are seated around a circular table in that order. They play a game of "countdown" where Henry starts by saying '34', and then each student says the next number in the countdown in their circular order. Who will eventually say '1'? Provide the name of the student.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{Fred}
There are  $27$  unit cubes. We are marking one point on each of the two opposing faces, two points on each of the other two opposing faces, and three points on each of the remaining two opposing faces of each cube. We are constructing a  $3\times 3 \times 3$  cube with these  $27$  cubes. What is the least number of marked points on the faces of the new cube?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{90}
Raman's salary was initially decreased by 40%, then increased by 20%, and finally decreased by 10%. What is the overall percentage change in his salary?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{35.2\%}
已知$p:{{\left( x-a \right)}^{2}}<9$，$q:{{\log }_{3}}\left( x+2 \right)<1$．若$\neg p$是$\neg q$的充分不必要条件，则$a$的取值范围是________．
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\left[ -2,1 \right]$}
4. Given $n$ positive integers $x_{1}, x_{2}, \cdots, x_{n}$ whose sum is 2016. If these $n$ numbers can be divided into 32 groups with equal sums and also into 63 groups with equal sums, find the minimum value of $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{94}
We placed 6 different dominoes in a closed chain on the table. The total number of points on the dominoes is $D$. What is the smallest possible value of $D$? (The number of points on each side of the dominoes ranges from 0 to 6, and the number of points must be the same on touching sides of the dominoes.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Alice refuses to sit next to either Bob or Carla. Derek refuses to sit next to Eric. Find the number of ways for the five of them to sit in a row of $5$ chairs under these conditions.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{28}
Two spheres are inscribed in a dihedral angle such that they touch each other. The radius of one sphere is 4 times that of the other, and the line connecting the centers of the spheres forms an angle of \(60^\circ\) with the edge of the dihedral angle. Find the measure of the dihedral angle. Provide the cosine of this angle, rounded to two decimal places if necessary.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.04}
The sum of \(\frac{1}{2 \times 3 \times 4}+\frac{1}{3 \times 4 \times 5}+\frac{1}{4 \times 5 \times 6}+\ldots+\frac{1}{13 \times 14 \times 15}+\frac{1}{14 \times 15 \times 16}\) is \(\frac{m}{n}\) in its lowest terms. Find the value of \(m+n\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{173}
For each positive integer $n$, let $f_1(n)$ be twice the number of positive integer divisors of $n$, and for $j \ge 2$, let $f_j(n) = f_1(f_{j-1}(n))$. For how many values of $n \le 50$ is $f_{50}(n) = 12?$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
In a television program, five children (Tian Tian, Shi Tou, Kimi, Cindy, Angela) need to switch fathers (each child can choose any one of the other four fathers except their own). Calculate the total number of different combinations of choices for the five children.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{44}
Find the number of eight-digit integers comprising the eight digits from 1 to 8 such that \( (i+1) \) does not immediately follow \( i \) for all \( i \) that runs from 1 to 7.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16687}
Find the number of real roots of the equation:
\[ 4 \cos (2007a) = 2007a. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Consider a regular hexagon $ABCDEF$. A frog starts at vertex $A$ and can jump to either of its two adjacent vertices on each move. If the frog reaches vertex $D$ within 5 jumps, it stops jumping. Otherwise, after making 5 jumps without reaching $D$, it also stops. How many different sequences of jumps can the frog make from the start until it stops?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{26}
In order to investigate the number of students from various classes participating in the extracurricular calligraphy group in a school, 5 classes were randomly selected, and the number of students from each class participating in the group was taken as a sample. It is known that the sample mean is 7, the sample variance is 4, and the sample data are all different. What is the maximum value among the sample data?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Let \((1+x+x^2)^n = a_0 + a_1x + a_2x^2 + \ldots + a_{2n}x^{2n}\) be an identity in \(x\). If we let \(s = a_0 + a_2 + a_4 + \ldots + a_{2n}\), the original answer is in the form \(\frac{k \cdot 3^n + m}{n}\), where n cannot be divisible by 2. Please find the value of \(k + m + n\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Increase Grisha's yield by 40% and Vasya's yield by 20%.

Grisha, the most astute among them, calculated that in the first case their total yield would increase by 1 kg; in the second case, it would decrease by 0.5 kg; in the third case, it would increase by 4 kg. What was the total yield of the friends (in kilograms) before their encounter with Hottabych?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
How many pairs (x, y) of non-negative integers with 0 <= x <= y satisfy the equation 5x^2 - 4xy + 2x + y^2 = 624? Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
The polynomial \( P(x) = x^{3} - 2x^{2} - x + 1 \) has the three real zeros \( a > b > c \). Find the value of the expression:
\[ a^{2} b + b^{2} c + c^{2} a. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Determine the domain of the function $f(x)= \dfrac {1}{ \sqrt {1-(\log _{2}(\cos x))^{2}}}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(2k\pi- \dfrac {\pi}{3},2k\pi+ \dfrac {\pi}{3})}
There are lily pads in a row numbered $0$ to $11$, in that order. There are predators on lily pads $3$ and $6$, and a morsel of food on lily pad $10$. Fiona the frog starts on pad $0$, and from any given lily pad, has a $\frac{1}{2}$ chance to hop to the next pad, and an equal chance to jump $2$ pads. What is the probability that Fiona reaches pad $10$ without landing on either pad $3$ or pad $6$? Express your answer as a fraction in simplest form in the format \frac{k}{m}, and find the value of k + m.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{271}
Evaluate the improper integral $\int_{0}^{\infty} \frac{\cos(x) - e^{-x}}{x} \, dx$ using advanced calculus or elementary real analysis methods without resorting to complex analysis.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Lucky Larry's teacher asked him to substitute numbers for $a$, $b$, $c$, $d$, and $e$ in the expression $a - (b - (c - (d + e)))$ and evaluate the result. Larry ignored the parentheses but added and subtracted correctly and obtained the correct result by coincidence. The numbers Larry substituted for $a$, $b$, $c$, and $d$ were $1$, $2$, $3$, and $4$, respectively. What number did Larry substitute for $e$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Jules is walking dogs to help pay for his family's vacation. There are five members in his family and each one must contribute the same amount towards the vacation cost. He charges $2 to start a walk and $1.25 per block. If he walks 20 dogs and has to walk them a total of 128 blocks, what is the total cost of the vacation?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$200}
Determine the force of water pressure on the wall of a lock, which is $20 \mathrm{~m}$ long and $5 \mathrm{~m}$ high (assuming the lock is filled with water to the top).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2.45 \times 10^6 \, \text{N}}
A company decides to increase the price of a product by 20%. If they aim to increase their total income by 10% despite the decrease in demand, by what proportion must the demand decrease to meet this new income goal?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{12}}
In $\triangle ABC$, the lengths of the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively. $M$ is a point on side $AB$ such that $\overrightarrow{CM} = \lambda \overrightarrow{MP} (\lambda \in R)$ and $\overrightarrow{MP} = \frac{\overrightarrow{CA}}{|\overrightarrow{CA}|\cos A} + \frac{\overrightarrow{CB}}{|\overrightarrow{CB}|\cos B}$. It is also given that $|\overrightarrow{CM}| = \frac{c}{2}$ and $a^2 + b^2 = 2\sqrt{2}ab$. Find the value of angle $C$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\pi}{4}}
Given the ellipse $\dfrac {x^{2}}{a^{2}}+ \dfrac {y^{2}}{b^{2}}=1 (a > b > 0)$, with the foci $F_1$ and $F_2$, let $P$ be a point on the ellipse where the magnitude of the projection of $\overrightarrow{F_1 F_2}$ onto $\overrightarrow{F_1 P}$ is exactly $|\overrightarrow{F_1 P}|$, and the angle between them is $\dfrac{\pi}{6}$. Calculate the eccentricity $e$ of the ellipse.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{3} - 1}
In the quadrilateral \(ABCD\), diagonals \(AC\) and \(BD\) intersect at point \(K\). Points \(L\) and \(M\) are the midpoints of sides \(BC\) and \(AD\), respectively. Segment \(LM\) contains point \(K\). The quadrilateral \(ABCD\) is such that a circle can be inscribed in it. Find the radius of this circle, given that \(AB = 3\), \(AC = \sqrt{13}\), and \(LK: KM = 1: 3\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3}{2}}
Given a sequence $\{a_n\}$ where $a_1=1$, and for any $m$, $n\in\mathbb{N}^*$, it holds that $a_{m+n}=a_m+a_n+mn$. Calculate the value of $\sum\limits_{i=1}^{2017} \dfrac {1}{a_i}$. Express your answer as a single fraction.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac {2017}{1009}}
Let $ABC$ be a triangle with $AB=5, BC=4$ and $AC=3$. Let $\mathcal{P}$ and $\mathcal{Q}$ be squares inside $ABC$ with disjoint interiors such that they both have one side lying on $AB$. Also, the two squares each have an edge lying on a common line perpendicular to $AB$, and $\mathcal{P}$ has one vertex on $AC$ and $\mathcal{Q}$ has one vertex on $BC$. Determine the minimum value of the sum of the areas of the two squares.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{144}{49}}
A *quadratic* number is a real root of the equations  $ax^2 + bx + c = 0$  where  $|a|,|b|,|c|\in\{1,2,\ldots,10\}$ . Find the smallest positive integer  $n$  for which at least one of the intervals  $$ \left(n-\dfrac{1}{3}, n\right)\quad \text{and}\quad\left(n, n+\dfrac{1}{3}\right) $$  does not contain any quadratic number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
A triangle is formed by selecting three lattice points (points with integer coordinates) at random with replacement within the square $[-99, 100] \times [-99, 100]$. The probability that the area of this triangle (which may be degenerate) is an integer, when expressed as the fraction $\frac{m}{n}$, where $m$ and $n$ are coprime positive integers, is needed. Calculate $m + n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
Kolya, after walking one-fourth of the way from home to school, realized that he forgot his problem book. If he does not go back for it, he will arrive at school 5 minutes before the bell rings, but if he goes back, he will be 1 minute late. How long (in minutes) does it take to get to school?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Given the set
\[
A=\left\{n \in \mathbf{Z} \left\lvert\,\left(1+\frac{1}{n}\right)^{n+1}=\left(1+\frac{1}{2020}\right)^{2000}\right.\right\} \text{,}
\]
determine the number of elements in the set \( A \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
For a sequence $\{a_{n}\}$ with $10$ terms, if $\{a_{n}\}$ satisfies $1\leqslant |a_{i+1}-a_{i}|\leqslant 2$ (where $i$ is a positive integer, $i\in \left[1,9\right]$), and $a_{1}=a_{10}\in \left[-1,0\right]$, let $k\in \{a_{n}|a_{n} \gt 0\}$. Find the maximum value of $k$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
In a house, a hall is of a certain length, 15 m wide, and 5 m high. Its interior has to be covered with mat. The total expenditure for covering the hall with mat at a cost of Rs. 40 per square meter is Rs. 38,000. What is the length of the hall?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32}
Two outlet pipes are connected to a cistern. The first outlet pipe can empty 3/4 of the cistern in 12 minutes, and the second can empty 1/2 of the cistern in 15 minutes. If both pipes are opened together and the first one operates for 4 minutes before being closed while the second one continues to operate for another 6 minutes, what part of the cistern will be emptied by the end of this process?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7/12}
Luis needs to ship a package of flour to a baker. He has $90$ cubic meters of flour to send. The post office will not ship any box which has an edge $10$ meters or longer. What are the dimensions of a box which will hold exactly $90$ cubic meters of flour, but whose edge lengths are all less than $10$ meters?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5 \times 3 \times 6}
Two circles \(C_{1}\) and \(C_{2}\) touch each other externally and the line \(l\) is a common tangent. The line \(m\) is parallel to \(l\) and touches the two circles \(C_{1}\) and \(C_{3}\). The three circles are mutually tangent. If the radius of \(C_{2}\) is 9 and the radius of \(C_{3}\) is 4, calculate the radius of \(C_{1}\). Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
A geometric sequence of positive integers starts with the first term as 5, and the fifth term of the sequence is 320. Determine the second term of this sequence.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Evaluate the sum: \( \frac{1}{a_1} + \frac{1}{a_2} + \ldots + \frac{1}{a_{20}} \), where \( a_n = \sqrt{1 + \left(1 - \frac{1}{n}\right)^2} + \sqrt{1 + \left(1 + \frac{1}{n}\right)^2} \) for \( n \ge 1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
For a given cube, count the number of "orthogonal line-plane pairs" where a straight line determined by two vertices is perpendicular to a plane containing four vertices of the cube.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36}
A right trapezoid has an upper base that is 60% of the lower base. If the upper base is increased by 24 meters, it becomes a square. What was the original area of the right trapezoid in square meters?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2880}
A sells a bicycle to B at a profit of 35% and B sells it to C at a loss of 25%. At this point, C gets a 10% discount on the purchase. Then C sells it to D at a profit of 20% and D finally sells it to E at a loss of 15%. Along the way, there is a 5% sales tax applied when B sells to C, and a 7% sales tax when D sells to E. Find the overall percentage of profit or loss on the original price, considering all the taxes and discounts in the transactions, when the bicycle was sold from A to E.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4.43\%}
Square $AXYZ$ is inscribed in equiangular hexagon $ABCDEF$ with $X$ on $\overline{BC}$, $Y$ on $\overline{DE}$, and $Z$ on $\overline{EF}$. Suppose that $AB=40$, and $EF=41(\sqrt{3}-1)$. What is the side-length of the square?

<asy>
size(200);
defaultpen(linewidth(1));
pair A=origin,B=(2.5,0),C=B+2.5*dir(60), D=C+1.75*dir(120),E=D-(3.19,0),F=E-1.8*dir(60);
pair X=waypoint(B--C,0.345),Z=rotate(90,A)*X,Y=rotate(90,Z)*A;
draw(A--B--C--D--E--F--cycle);
draw(A--X--Y--Z--cycle,linewidth(0.9)+linetype("2 2"));
dot("$A$",A,W,linewidth(4));
dot("$B$",B,dir(0),linewidth(4));
dot("$C$",C,dir(0),linewidth(4));
dot("$D$",D,dir(20),linewidth(4));
dot("$E$",E,dir(100),linewidth(4));
dot("$F$",F,W,linewidth(4));
dot("$X$",X,dir(0),linewidth(4));
dot("$Y$",Y,N,linewidth(4));
dot("$Z$",Z,W,linewidth(4));
</asy>
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{29\sqrt{3}}
在 $\triangle A B C$ 中, $O$ 为 $\triangle A B C$ 的外心，且满足 $\overrightarrow{A O} \cdot \overrightarrow{A B}+\overrightarrow{B O} \cdot \overrightarrow{B C}=\overrightarrow{C O} \cdot \overrightarrow{C A}$. 计算$\angle B$ 的大小，答案形式为 \frac{m\pi}{n}，请给出 m + n 的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Some bugs are sitting on squares of  $10\times 10$  board. Each bug has a direction associated with it **(up, down, left, right)**. After 1 second, the bugs jump one square in **their associated**direction. When the bug reaches the edge of the board, the associated direction reverses (up becomes down, left becomes right, down becomes up, and right becomes left) and the bug moves in that direction. It is observed that it is **never** the case that two bugs are on same square. What is the maximum number of bugs possible on the board?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40}
Find the value of $\dfrac{2\cos 10^\circ - \sin 20^\circ }{\sin 70^\circ }$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{3}}
A data with four real numbers $2n-4$, $2n-6$, $n^2-8$, $3n^2-6$ has an average of $0$ and a median of $9/2$. The largest number of such data is ...
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21}
In the image, there is a cube with an edge length of $2 \text{ cm}$ composed of eight smaller cubes, each with an edge length of $1 \text{ cm}$. Eight faces of the small cubes are painted black, while the rest are white. It is possible to assemble the bigger cube such that its surface is entirely white. In how many ways can the small cubes be painted? Assume that identically painted small cubes cannot be distinguished and can be interchanged.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
A regular hexagon $S$ has an inscribed triangle $H$ such that one side of $H$ is parallel to one side of $S$. What is the maximum possible ratio of the area of $H$ to the area of $S$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3}{8}}
Given an arithmetic sequence $\{a_n\}$ with a common difference $d \neq 0$, and $a_1=1$, $a_3$, $a_{13}$ form a geometric sequence. Find the minimum value of $\frac{2S_n+8}{a_n+3}$ for all positive integers $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{5}{2}}
During breaks, schoolchildren played table tennis. Any two schoolchildren played no more than one game against each other. At the end of the week, it turned out that Petya played half, Kolya - a third, and Vasya - one fifth of the total number of games played during the week. What could be the total number of games played during the week if it is known that at least two games did not involve Vasya, Petya, or Kolya?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
Find the number of lines that are tangent to the circle $(x^{2}+(y-2)^{2}=1)$ and have equal intercepts on both coordinate axes.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
A group of adventurers is showing their loot. It is known that exactly 5 adventurers have rubies; exactly 11 have emeralds; exactly 10 have sapphires; exactly 6 have diamonds. Additionally, it is known that:

- If an adventurer has diamonds, then they have either emeralds or sapphires (but not both simultaneously);
- If an adventurer has emeralds, then they have either rubies or diamonds (but not both simultaneously).

What is the minimum number of adventurers that can be in such a group?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
Five positive integers (not necessarily all different) are written on five cards. Boris calculates the sum of the numbers on every pair of cards. He obtains only three different totals: 57, 70, and 83. What is the largest integer on any card? Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{48}
Select 5 different letters from the word "equation" to arrange in a row, including the condition that the letters "qu" are together and in the same order.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{480}
Sienna gave Bailey half of her suckers. Jen ate 11 suckers and gave the rest to Molly. Molly ate some suckers and gave the rest to Harmony. Harmony kept 3 and passed the remainder to Taylor. Taylor ate one and gave the last 5 to Callie. How many suckers did Molly eat?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
Let $E$ and $F$ be the respective midpoints of $BC$ and $CD$ in a convex quadrilateral $ABCD$. Segments $AE$, $AF$, and $EF$ divide the quadrilateral into four triangles whose areas are four consecutive integers. Find the maximum possible area of $\Delta BAD$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Given that $x$ and $y$ satisfy $\log_{2}\left[4\cos^{2}(xy) + \frac{1}{4\cos^{2}(xy)}\right] = -y^2 + 4y - 3$, find the value of $y\cos{4x}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
将 $2018$ 行 $4$ 列方格表的每个格染成红色或蓝色，使得方格表每行的红格数与蓝格数相等，每列的红格数与蓝格数也相等。记 $m$ 为满足上述要求的染色方法的总数。求 $m$除以 $2018$ 所得的余数。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Find the value of $m$ such that \( \arg\left(\frac{(z_1^2z_2)^m}{3i}\right)=\frac{\pi}{2} \), where \( z_1=2e^{i\pi/3} \) and \( z_2=2e^{i2\pi/3} \), and \( \arg(z) \) is in the interval \([0,2\pi)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{3}{4}}