inclusionAI/Ring-lite-rl-data huggingface dataset:
Let $X$ be a normal random variable with mean $0$ and variance $1$. Let $Y$ be a Bernoulli random variable with $p = \frac{1}{2}$, and suppose that $X$ and $Y$ are independent. Define $Z = (Y + 1)X$. Calculate the conditional probability $P(Y = 1 \mid Z = 3)$. Provide your answer as a decimal rounded to three decimal places.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.936}
Evaluate the contour integral \( \oint \frac{z+2}{z^2+9} \, dz \) using the residue at infinity.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\pi i}
Let $x_5$ be the remainder when the positive integer $x$ is divided by 5. Compute the sum of all elements in the solution set of the equation: $$x^5(x_5)^5 - x^6 - (x_5)^6 + x(x_5) = 0.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1300}
Determine the order of the group $G$ with the presentation $\langle a,b \mid a^{16}=b^6=1,bab^{-1}=a^3\rangle$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{48}
Determine the smallest positive integer \( a \) such that \( x^4 + a^2 \) is not a prime number for any integer \( x \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
In trapezoid \(ABCD\), the bases \(AB\) and \(CD\) measure 55 and 31 units, respectively. The diagonals of the trapezoid are perpendicular to each other. Calculate the dot product of vectors \(\overrightarrow{AD}\) and \(\overrightarrow{BC}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1705}
Evaluate the limit \( \lim_{\textbf{X} \rightarrow \infty} \left(\sum_i (\ln(X_i))^a\right)^{1/a} \) where \( a \) is a negative constant. Determine whether the limit is 0 or infinity.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
9. (10 points) On a plane, there are 5 different lines, and these 5 lines form $m$ intersection points. How many different values can $m$ have?

保留源文本的换行和格式，翻译结果如下：

9. (10 points) On a plane, there are 5 different lines, and these 5 lines form $m$ intersection points. How many different values can $m$ have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
On a sphere of radius 1, there are four non-coplanar points A, B, C, D, and AB = CD = x, BC = DA = y, CA = BD = z. Calculate the value of x^2 + y^2 + z^2. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
What is the minimum number of points that must be marked inside a convex $n$-gon so that each triangle with vertices at the vertices of this $n$-gon contains at least one marked point?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n-2}
已知一个正三棱柱的各条棱长均为 $3$, 则其外接球的体积的$\frac{2 \sqrt{21}}{\pi}$为 $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{147}
a ring x of diameter 16 inches is placed inside another ring y of diameter of 18 inches . how much fraction of the ring x surface is not covered by the ring y ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Let  $E(n)$  denote the largest integer  $k$  such that  $5^k$  divides  $1^{1}\cdot 2^{2} \cdot 3^{3} \cdot \ldots \cdot n^{n}.$  Calculate  $$ \lim_{n\to \infty} \frac{E(n)}{n^2 }. $$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{8}}
Given that the derivative of a function f(x) is f'(x), if for any x in R, f'(x) - f(x) < 1, and f(0) = 2022, find the solution set of the inequality f(x) + 1 > 2023e^x.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-\infty, 0)}
How many paths are there from the starting point $C$ to the end point $D$, if every step must be up or to the right in a grid of 8 columns and 7 rows?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6435}
Given the sets $A=\{x|-1<x<3\}$ and $B=\{x|-1<x<m+1\}$. If $x \in A$ is a necessary but not sufficient condition for $x \in B$, then the range of the real number $m$ is \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-2, 2)}
An exam paper contains 4 multiple-choice questions, each with three options (A), (B), and (C). Several students take the exam, and it is found that for any group of 3 students, there is always at least one question where their answers are all different. Furthermore, each student answers every question. What is the maximum number of students who can take the exam?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Given a sequence $\left\{a_{n}\right\}$ where all terms are non-negative real numbers, and it satisfies: for any integer $n \geq 2$, $a_{n+1} = a_{n} - a_{n-1} + n$. If $a_{2} a_{2022} = 1$, find the maximum possible value of $a_{1}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{4051}{2025}}
A circular road runs around a circular garden. The sum of the circumferences of the inner and outer circles is 88 meters, and the radius of the inner circle is one-third of the outer circle. What is the width of the road?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
9.6. Waiting for customers, a watermelon seller sequentially weighed 20 watermelons (weighing 1 kg, 2 kg, 3 kg, ..., 20 kg), balancing the watermelon on one scale pan with one or two weights on the other pan (possibly identical). In the process, the seller wrote down on a piece of paper the weights he used. What is the smallest number of different numbers that could have appeared in his notes, if the weight of each weight is an integer number of kilograms?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Find the area of a right triangle where one leg is $1/3$ longer than the other leg and $1/3$ shorter than the hypotenuse.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2}{3}}
Draw a rectangle. Connect the midpoints of the opposite sides to get 4 congruent rectangles. Connect the midpoints of the lower right rectangle for a total of 7 rectangles. Repeat this process infinitely. Let $n$ be the minimum number of colors we can assign to the rectangles so that no two rectangles sharing an edge have the same color and $m$ be the minimum number of colors we can assign to the rectangles so that no two rectangles sharing a corner have the same color. Find the ordered pair $(n, m)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(3,4)}
Whole numbers that read the same from left to right and right to left are called symmetrical. For example, the number 513315 is symmetrical, whereas 513325 is not. How many six-digit symmetrical numbers exist such that adding 110 to them leaves them symmetrical?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{81}
A rotating beacon is located 2 miles out in the water. Let $A$ be the point on the shore that is closest to the beacon. As the beacon rotates at $10 \mathrm{rev} / \mathrm{min}$, the beam of light sweeps down the shore once each time it revolves. Assume that the shore is straight. Find the speed at which the point where the beam hits the shore is moving when the beam is lighting up a point 2 miles along the shore from the point $A$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$80 \pi \mathrm{mi} / \mathrm{min}$}
On Monday, Millie puts a quart of seeds, $25\%$ of which are millet, into a bird feeder. On each successive day she adds another quart of the same mix of seeds without removing any seeds that are left. Each day the birds eat only $25\%$ of the millet in the feeder, but they eat all of the other seeds. On which day, just after Millie has placed the seeds, will the birds find that more than half the seeds in the feeder are millet? If Monday is day 1, what is the number of the day?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
All the diagonals of a convex heptagon divide it into several non-overlapping smaller convex polygons. What is the maximum possible number of sides of these smaller polygons? Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
How many decimal places of accuracy does the following approximate equality hold?

$$
495 \sqrt{2}-388 \sqrt{3} \approx 28
$$

that is, how many zeros can we write after 28 following the decimal point, so that the resulting number is still the rounded value of the correct left-hand side?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
若函数 $f(x)=-\frac{1}{2} x^{2}+\frac{13}{2}$ 在区间 $[a, b]$ 上的最小值为 $2 a$, 最大值为 $2 b$, 求 $[a, b]$. 标准答案格式为$[a,b]=[m,n]$ 或 $\Big[-a-\sqrt{b},\frac{c}{d}\Big]$，请提供区间 $[a, b]$ 的端点m+n+a+b+c+d和。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40}
Let \( n \geq 3 \) be an integer. What is the largest possible number of interior angles greater than \( 180^\circ \) in an \( n \)-gon in the plane, given that the \( n \)-gon does not intersect itself and all its sides have the same length?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n-3}
A multiple choice test consists of 100 questions. If a student answers a question correctly, he will get 4 marks; if he answers a question wrongly, he will get \(-1\) mark. He will get 0 mark for an unanswered question. Determine the number of different total marks of the test. (A total mark can be negative.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{495}
Compute
\[\cos^6 0^\circ + \cos^6 1^\circ + \cos^6 2^\circ + \dots + \cos^6 90^\circ.\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{229}{8}}
一个 $10 \times 10$ 的数表在初始状态下每个格内均填有数字1，现在对数表进行如下操作：选择某一行或某一列，并将该行（列）中所有格子内的数变为其相反数。对于非负整数 N ，如果可经过有限次操作，使得最终数表中 -1 的个数恰为 N ，则称 N 是 "好的"，则好的 N 的个数为 $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{29}
Find the greatest real number  $k$  such that, for any positive  $a,b,c$  with  $a^{2}>bc$ ,  $(a^{2}-bc)^{2}>k(b^{2}-ca)(c^{2}-ab)$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Josh gets together with 12 friends. Everyone including him puts 5 dollars into a pot. First place gets 65% of the money, while second and third place split the remaining 35%. How much money does third place get?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$11.38}
For what smallest natural number \( n \) can the numbers from 1 to \( n \) be arranged in a circle such that each number is either greater than all of the 40 numbers following it in the clockwise direction or smaller than all of the 30 numbers following it in the clockwise direction?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{70}
A square $ABCD$ with side length 4 is folded along $BD$ to form a dihedral angle of $60^{\circ}$, then the distance between the midpoint of $BC$ and point $A$ is $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\sqrt{2}}
The sequence $\{a_n\}$ satisfies $a_{n+1} + (-1)^n a_n = 2n - 1$. Find the sum of the first 60 terms of $\{a_n\}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1830}
Given the hyperbola $C$: $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1 (a > 0, b > 0)$ with its right focus at $F(c,0)$, points $M$ and $N$ are on hyperbola $C$, $O$ is the coordinate origin. If quadrilateral $OFMN$ is a parallelogram with an area of $bc$, determine the eccentricity of hyperbola $C$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\sqrt{2}}
Given two lines $l_1: y = x + \frac{1}{2}a$ and $l_2: y = (a^2-3)x + 1$, if $l_1 \parallel l_2$, then the value of $a$ is
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2}
两个圆内切于 $K$, 大圆的弦 $A B$ 与小圆切于 $L$, 已知 $A K: B K=2: 5, A L=10$ ，求 $B L$ 的长。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25}
Find the square root of $\dfrac{10!}{210}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24\sqrt{30}}
What is the number of different triangles that can be obtained when each side of a square is divided into 8 equal parts, and using the divisions as vertices (not including the vertices of the square)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3136}
There are 37 students who need to cross a river. There is a small boat at the ferry that can carry up to 5 people. To get all of them across the river, what is the minimum number of times the boat must be used? Note that a round trip counts as one time. Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Given the vector equation $\overrightarrow{OA} + \overrightarrow{OB} + \sqrt{2}\overrightarrow{OC} = \overrightarrow{0}$, where O is the circumcenter of triangle ABC, find the exact measure of angle C in radians.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\pi}{4}}
Given that line $l$: $x+ay-1=0 (a\in R)$ is the symmetric axis of circle $C$: $x^2+y^2-4x-2y+1=0$, and it passes through point $A(-4,a)$. Draw a tangent line from point $A$ to circle $C$, with the tangent point being $B$. Calculate the length of $|AB|$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Let  $a$  be a positive real number,  $n$  a positive integer, and define the *power tower*  $a\uparrow n$  recursively with  $a\uparrow 1=a$ , and  $a\uparrow(i+1)=a^{a\uparrow i}$  for  $i=1,2,3,\ldots$ .  For example, we have  $4\uparrow 3=4^{(4^4)}=4^{256}$ , a number which has  $155$  digits.  For each positive integer  $k$ , let  $x_k$  denote the unique positive real number solution of the equation  $x\uparrow k=10\uparrow (k+1)$ .  Which is larger:  $x_{42}$  or  $x_{43}$ ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ x_{42} }
In a sign pyramid, a cell gets a "+" if the two cells below it have the same sign, and it gets a "-" if the two cells below it have different signs. The pyramid has four levels, and the bottom row has four cells. How many possible ways are there to fill the four cells in the bottom row to produce a "+" at the top of the pyramid? Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Triangle $ABC$ has side lengths $AB=231$, $BC=160$, and $AC=281$. Point $D$ is constructed on the opposite side of line $AC$ as point $B$ such that $AD=178$ and $CD=153$. Compute the distance from $B$ to the midpoint of segment $AD$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{208}
Given the wins of five baseball teams as displayed in the graph, identify the number of games won by the Patriots. The clues are as follows:
1. The Tigers won more than the Eagles.
2. The Patriots won more than the Cubs, but fewer than the Mounties.
3. The Cubs won more than 15 games.
4. The Falcons won more games than the Eagles but fewer than the Mounties.

How many games did the Patriots win? 

Graph data:
Teams (in unknown order): 10, 18, 22, 27, 33 games.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{27}
At the school, there are three times as many girls as boys in the Biology class. The Physics class has a 2:3 ratio of boys to girls. The Chemistry class has 270 students, with an equal number of boys and girls. If the Biology class has half as many students as the Physics class, and the total number of students in all three classes is 1000, how many boys are in the Biology class?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60}
Let there be a scalene triangle $ABC$, and denote $M$ by the midpoint of $BC$. The perpendicular bisector of $BC$ meets the circumcircle of $ABC$ at point $P$, on the same side with $A$ with respect to $BC$. Let the incenters of $ABM$ and $AMC$ be $I,J$, respectively. Let $\angle BAC=\alpha$, $\angle ABC=\beta$, $\angle BCA=\gamma$. Find $\angle IPJ$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\alpha}{2}}
## Task 12/78

When asked how old he is, someone answers that in the year $x^{2}$ he was exactly $x$ years old. In which year was he born?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1892}
Given the binomial series expansion of $\displaystyle (1-x)^{\frac{1}{2}}$ as $a_{0}+a_{1}x+a_{2}x^2+\cdots$ and the condition that $a_{0}+a_{1}+a_{2}+\cdots+a_{10}=\frac{\binom{20}{10}}{k^{10}}$, find the value of $k$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
0 < \(a_1 \le a_2 \le a_3 \le \ldots\) is an unbounded sequence of integers. Let \(b_n = m\) if \(a_m\) is the first member of the sequence to equal or exceed \(n\). Given that \(a_{19} = 85\), what is the maximum possible value of \(a_1 + a_2 + \ldots + a_{19} + b_1 + b_2 + \ldots + b_{85}\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1700}
已知椭圆$C:\frac{{{x}^{2}}}{8}+\frac{{{y2}}}{2}=1$与圆$M:{{x}^{2}}+{{y2}}+2\sqrt{2}+2-{{r}^{2}}=0(0< r<\sqrt{2})$，过椭圆$C$的上顶点$P$作圆$M$的两条切线分别与椭圆$C$相交于$A,B$两点（不同于$P$点），则直线$PA$与直线$PB$的斜率之积等于__________．
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Given a moving line $l$ that tangentially touches the circle $O: x^{2}+y^{2}=1$ and intersects the ellipse $\frac{x^{2}}{9}+y^{2}=1$ at two distinct points $A$ and $B$, find the maximum distance from the origin to the perpendicular bisector of line segment $AB$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{4}{3}}
The sum of all solutions to the equation \( 2(x-1) \sin \pi x + 1 = 0 \) in the interval \([-2, 4]\) is equal to _____.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Given that the optimal amount of a material to add lies between 100g and 1100g, if we use the 0.618 method to arrange the experiment, and the first and second test points are $x_1$ and $x_2$ ($x_1 > x_2$) respectively, then if $x_2$ is the better point, the third test point $x_3$ should be ____ g (Please answer with a number).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{336\text{g}}
In an isosceles right triangle \( \triangle ABC \), \( \angle A = 90^\circ \), \( AB = 1 \). \( D \) is the midpoint of \( BC \), \( E \) and \( F \) are two other points on \( BC \). \( M \) is the other intersection point of the circumcircles of \( \triangle ADE \) and \( \triangle ABF \); \( N \) is the other intersection point of line \( AF \) with the circumcircle of \( \triangle ACE \); \( P \) is the other intersection point of line \( AD \) with the circumcircle of \( \triangle AMN \). Find the length of \( AP \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{2}}
During vacation, Jimmy visits different beaches on an island, so he changes his place of lodging from time to time. The first 3 days he stays in a hostel, where he is charged $15 per night. However, there is a 10% service tax that is added to the nightly charge. The fourth and fifth days he stays in a cabin where he shares expenses with 2 of his friends, and they are charged $45 total per night, with a 7% government tax charged on top of the base cost. On the third day, Jimmy gets a discount code from a fellow traveler, which gives him a 15% discount on the cabin for the 5th day. How much did Jimmy spend on lodging during his vacation, including taxes and considering the discount he received?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$79.19}
Four cousins Alan, Bob, Carl, and Dan are 3, 8, 12, and 14 years old, although not necessarily in that order. Alan is younger than Carl. The sum of the ages of Alan and Dan is divisible by 5. The sum of the ages of Carl and Dan is divisible by 5. What is the sum of the ages of Alan and Bob?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{17}
If for a natural number \( n (\geqslant 2) \), there exist integers \( a_{1}, a_{2}, \cdots, a_{n} \) such that
\[ 
a_{1}+a_{2}+\cdots+a_{n}=a_{1} \cdot a_{2} \cdot \cdots \cdot a_{n}=1990,
\]
find the minimum value of \( n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Given a sequence $\{a_n\}$ satisfying $a_1=1$, $a_2=2$, and $\frac{1}{2}na_{n+2}$ is the arithmetic mean of $(n+2)a_n$ and $\lambda (n^2+2n)$, if the sequence $\{a_n\}$ is increasing, then the range of the real number $\lambda$ is ______________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[0,+\infty)}
已知函数 $y=f(x)$ 的图像既关于点 $(1,1)$ 中心对称, 又关于直线 $x+y=0$轴对称。若 $x \in(0,1)$ 时, $f(x)=\log _{2}(x+1)$, 则 $f\left(\log _{2} 10\right)$ 的值为 $\qquad$.
原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{22}
Let the function $f(x)=A\sin(\omega x+\varphi)$ ($A$, $\omega$, $\varphi$ are constants, $A > 0$, $\omega > 0$). If $f(x)$ is monotonic in the interval $\left[\frac{\pi}{6}, \frac{\pi}{2}\right]$ and $f\left(\frac{\pi}{2}\right)=f\left(\frac{2\pi}{3}\right)=-f\left(\frac{\pi}{6}\right)$, then the smallest positive period of $f(x)$ is \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi}
Let $ABC$ be a right triangle with $m(\widehat{A})=90^\circ$. Let $APQR$ be a square with area $9$ such that $P \in [AC]$, $Q \in [BC]$, $R \in [AB]$. Let $KLMN$ be a square with area $8$ such that $N,K \in [BC]$, $M \in [AB]$, and $L \in [AC]$. Find $|AB|+|AC|$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
You are given that \[17! = 355687ab8096000\] for some digits $a$ and $b$. Find the two-digit number $\overline{ab}$ that is missing above.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{42}
Two trains of different lengths run at the speed of 36 kmph and 18 kmph in opposite directions in parallel tracks. The time which they take to cross each other is 46.66293363197611 seconds. If the other train is 400 m long, how long is the first train?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{300}
Find the minimum number of solutions of $(k'(x))^2+k(x)k''(x)=0$ in $[a,e]$ for a twice differentiable function $k:\Bbb R\to\Bbb R$ such that $ k(a)=k(b)=0 ,k(c)=k(d)=1 , k(e)=2$, where $a<b<c<d<e$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
设 $O, A, B, C$ 为空间内四点, $\overrightarrow{O A}=\boldsymbol{a}, \overrightarrow{O B}=\boldsymbol{b}, \overrightarrow{O C}=\boldsymbol{c}$, 且 $\boldsymbol{a}+\boldsymbol{b}+\boldsymbol{c}=\mathbf{a}, \boldsymbol{a} \cdot \boldsymbol{b}=\boldsymbol{b} \cdot \boldsymbol{c}=\boldsymbol{c} \cdot \boldsymbol{a}$ $=-1$, 则 $|\boldsymbol{a}|+|\boldsymbol{b}|+|\boldsymbol{c}|=$ $\qquad$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$3 \sqrt{2}$}
What are the spherical coordinates of point M with the Cartesian coordinates $(-1, -1, \sqrt{2})$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(2, \frac{\pi}{4}, \frac{5\pi}{4})}