
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 left and right foci of the hyperbola $\frac{x^2}{9} - \frac{y^2}{4} = 1$, and point $P$ is on the hyperbola such that $\overrightarrow{PF_1} \cdot \overrightarrow{PF_2} = 0$, find the value of $|\overrightarrow{PF_1} + \overrightarrow{PF_2}|$.

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