
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
(1) Given vectors $\overrightarrow{a}=(1,-1)$ and $\overrightarrow{b}=(x,1)$, and $\overrightarrow{a}\parallel \overrightarrow{b}$, then $x=\boxed{-1}$.

(2) Point $P$ is any point inside a cube $ABCD-A_{1}B_{1}C_{1}D_{1}$ with side length $2$. The probability that the distance from point $P$ to point $A$ is less than $1$ is $\boxed{\frac{\pi}{48}}$.

(3) In $\triangle ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given $c \cos B+(b-4a)\cos C=0$ and $a=1$, $b=2$, then the area of $\triangle ABC$ is $\boxed{\frac{\sqrt{15}}{4}}$.

(4) The angle between vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ is $\frac{\pi }{3}$, $|\overrightarrow{b}|=2$ and $(\overrightarrow{a}+2\overrightarrow{b})\cdot (\overrightarrow{a}-\overrightarrow{b})=-2$. The projection of $\overrightarrow{a}$ in the direction of $\overrightarrow{b}$ is $\boxed{1}$.

Present the answer in LaTex format: \boxed{Your answer}
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