
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 equation of the parabola $C_1$ is $y^2 = 4x$, and the focus $F$ is the right focus of the hyperbola $C_2: \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$. Point $P$ is a common point of curves $C_1$ and $C_2$, and point $M$ is on the directrix of parabola $C_1$. If $\triangle FPM$ is an isosceles right triangle with $P$ as the right angle vertex, find the eccentricity of the hyperbola $C_2$.

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