
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
The right focus of the hyperbola $C:\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\left(a \gt 0,b \gt 0\right)$ is $F$. A line passing through point $F$ intersects the right branch of the hyperbola $C$ at points $A$ and $B$. Point $A$ is symmetric to the origin $O$ with respect to point $P$, $PF\perp AB$, and $|BF|=3|AF|$. Find the eccentricity of the hyperbola $C$.

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