
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) If the proposition "$\forall x \in \left[-\frac{\pi}{4}, \frac{\pi}{3}\right], \tan x \leqslant m$" is true, then the minimum value of the real number $m$ is \_\_\_\_\_\_\_\_\_\_\_\_.

(2) Given that $F_1$ and $F_2$ are the two foci of the ellipse $\frac{x^2}{25} + \frac{y^2}{9} = 1$, a line passing through $F_1$ intersects the ellipse at points $A$ and $B$. If $|F_2A| + |F_2B| = 13$, then $|AB| = $ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_.

(3) The length of the real axis of the hyperbola $y^2 - 2x^2 = -8$ is \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_.

(4) Given that point $P(-1,3)$ is on the directrix of the parabola $C: y^2 = 2px$, with focus $F$, then the slope of line $PF$ is \_\_\_\_\_\_\_\_\_\_.

(5) Given that the ellipse $\frac{x^2}{25} + \frac{y^2}{16} = 1$ and the hyperbola $\frac{x^2}{m} - \frac{y^2}{8} = 1$ share the same foci $F_1, F_2$, and one intersection point is $P$, then the value of $\overrightarrow{PF_1} \cdot \overrightarrow{PF_2}$ is \_\_\_\_\_\_\_\_\_\_\_\_\_\_.

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