
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 a random variable $X$ that takes the values $1, 2, 3, \ldots, n$ with equal probability, if $P(X < 5)=0.2$, then the value of $n$ is ______.

(2) In the cube $ABCD-A_{1}B_{1}C_{1}D_{1}$, the sine of the angle formed by the line $BC_{1}$ and the plane $A_{1}BD$ is ______.

(3) Given that $F$ is the left focus of the hyperbola $\frac{x^{2}}{4}- \frac{y^{2}}{12}=1$, $A(1,\sqrt{7})$, and $P$ is a moving point on the right branch of the hyperbola, the minimum value of $|PF|+|PA|$ is ______.

(4) Given the function $f(x)=\begin{cases} x^{2}+2x+a, & x < 0, \\ \ln x, & x > 0, \end{cases}$ where $a$ is a real number, and $A(x_{1},f(x_{1}))$, $B(x_{2},f(x_{2}))$ are two points on the graph of the function with $x_{1} < x_{2}$. If the tangents to the graph of $f(x)$ at points $A$ and $B$ coincide, then the range of values for the real number $a$ is ______.

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