
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 $\frac{x^2}{9} - \frac{y^2}{4} = 1$, respectively, and $A$ is a point on the right branch of the hyperbola. If $2\overrightarrow{OP} = \overrightarrow{OA} + \overrightarrow{OF_1}$ and $2\overrightarrow{OQ} = \overrightarrow{OA} + \overrightarrow{OF_2}$, then $|\overrightarrow{OQ}| - |\overrightarrow{OP}| = \_\_\_\_\_\_\_\_.$

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