
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 a hyperbola $C$: $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ $(a>0, b>0)$ with left and right foci $F_{1}$, $F_{2}$, and a point $P$ on $C$, where $\angle F_{1}PF_{2}=60^{\circ}$, find the eccentricity $e$ of $C$ when $\sin \angle PF_{2}F_{1}=3\sin \angle PF_{1}F_{2}$.

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