
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 point \( P \) is on the ellipse \( \frac{x^2}{25} + \frac{y^2}{9} = 1 \), and \( F_1 \) is its left focus, \( Q \) lies on \( PF_1 \) and satisfies the condition
$$
\overrightarrow{O Q} = \frac{1}{2} \left( \overrightarrow{O P} + \overrightarrow{O F_1} \right), \quad |\overrightarrow{O Q}| = 3.
$$

Find the distance from point \( P \) to the left directrix of the ellipse.

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