
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 point $P$ on the right branch of the hyperbola $C$: $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ ($a > 0, b > 0$), and $F_1$, $F_2$ are the left and right foci of the hyperbola, respectively. It is also given that $|\vec{PF_1} + \vec{PF_2}| = 2c$, and the area of $\triangle PF_1F_2$ is $ac$. Find the eccentricity of the hyperbola.

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