
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 an ellipse $E$ with its left and right foci as $F_{1}$ and $F_{2}$, respectively. A line passing through $F_{1}$ with a slope of $2$ intersects the ellipse $E$ at points $P$ and $Q$. If $\triangle PF_{1}F_{2}$ is a right-angled triangle, then the eccentricity of the ellipse $E$ is ______.

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