
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
Given that $F\_1$ and $F\_2$ are the two foci of the hyperbola $C$: $\frac{x^{2}}{9} - \frac{y^{2}}{m} = 1$, and point $P$ is on $C$ such that $\overrightarrow{PF_{1}} \cdot \overrightarrow{PF_{2}} = 0$. If the directrix of the parabola $y^{2} = 16x$ passes through a focus of the hyperbola $C$, find the value of $|\overrightarrow{PF_{1}}| \cdot |\overrightarrow{PF_{2}}|$.

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