
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
Let $F_1$ and $F_2$ be the left and right foci of the hyperbola $C: \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ ($a > 0$, $b > 0$), respectively. Let $P$ be a point on the right branch of $C$, and let the ray $PT$ bisect $\angle F_1PF_2$. Draw a parallel line to $PT$ through the origin $O$, intersecting $PF_1$ at point $M$. If $|MP| = \frac{1}{3}|F_1F_2|, calculate the eccentricity of $C$.

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