
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 negation of the proposition $\exists x_0 < 0, x_0^2 > 0$ is __________.

(2) If there is a point $M$ on the parabola $y^2= -2px (p > 0)$ with an abscissa of $-9$ and its distance to the focus is $10$, then the coordinates of point $M$ are __________.

(3) In $\Delta ABC$, the sides opposite to angles $A, B, C$ are $a, b, c$ respectively. If $\sin A = \sqrt{3} \sin C, B = 30^\circ, b = 2$, then the length of side $c$ is __________.

(4) Given $F$ is the left focus of the hyperbola $C: \frac{x^2}{4} - \frac{y^2}{12} = 1$, $A(1,4)$, and $P$ is a point on the right branch of $C$. When the perimeter of $\triangle APF$ is minimized, the distance from point $F$ to line $AP$ is __________.

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