
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 the hyperbola $$\frac{x^2}{16} - \frac{y^2}{48} = 1$$ with left and right foci $F_1$ and $F_2$, respectively, let $P$ be a point on C such that $\overrightarrow{F_1Q} = \overrightarrow{QP}$, where $O$ is the coordinate origin. If $|PF_1| = 10$, then find $|OQ|$.

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