
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. The converse of the proposition "If $x > 1$, then ${x}^{2} > 1$" is ________.
2. Let $P$ be a point on the parabola ${{y}^{2}=4x}$ such that the distance from $P$ to the line $x+2=0$ is $6$. The distance from $P$ to the focus $F$ of the parabola is ________.
3. In a geometric sequence $\\{a\_{n}\\}$, if $a\_{3}$ and $a\_{15}$ are roots of the equation $x^{2}-6x+8=0$, then $\frac{{a}\_{1}{a}\_{17}}{{a}\_{9}} =$ ________.
4. Let $F$ be the left focus of the hyperbola $C$: $\frac{{x}^{2}}{4}-\frac{{y}^{2}}{12} =1$. Let $A(1,4)$ and $P$ be a point on the right branch of $C$. When the perimeter of $\triangle APF$ is minimum, the distance from $F$ to the line $AP$ is ________.

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