
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 an ellipse $\frac{x^2}{25} + \frac{y^2}{16} = 1$, let point P on the ellipse be such that the distance from P to the left directrix is 10. Let F be the left focus of the ellipse, and let M satisfy $\vec{OM} = \frac{1}{2}(\vec{OP} + \vec{OF})$. Determine the length of $\vec{OM}$.

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