
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 the ellipse with foci F1 and F2 described by $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ ($a > b > 0$), and $\vec{PF_1} \cdot \vec{PF_2} = 0$ and $\tan ∠PF_1F_2 = \frac{1}{2}$, calculate the eccentricity of this ellipse.

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