
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 $\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1\left(a \gt b \gt 0\right)$ with left and right foci $F_{1}$ and $F_{2}$, point $P$ is a point on the ellipse, and line segment $F_{1}P$ intersects the $y$-axis at point $Q$. If $|PQ|=2|QF_{1}|$ and triangle $\triangle PF_{1}F_{2}$ is an isosceles triangle, then the eccentricity of the ellipse is ____.

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