
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 two foci of the ellipse $$C: \frac{x^2}{36} + \frac{y^2}{27} = 1$$, and point $P$ is a point on ellipse $C$ with $|PF_1| = 8$. If $M$ is the midpoint of the line segment $PF_1$, find the length of the line segment $OM$.

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