
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 points $F_1$ and $F_2$ are the left and right foci of the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \ (a > 0, b > 0)$, respectively. Let point $P$ be a point on the right branch of the hyperbola. If the x-coordinate of point $P$ is $x_0 = \frac{4}{3}a$ when $F_1P \perp F_2P$, find the eccentricity $e$ of the hyperbola.

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