
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 $C:\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1(a>b>0)$ with two foci $F_{1}$ and $F_{2}$. Points $P$ and $Q$ are two points on $C$ symmetric with respect to the origin, and $|PQ|=|F_{1}F_{2}|$. If the area of $\triangle PF_{2}Q$ is $S≥\frac{1}{8}|PQ|^{2}$, then the range of the eccentricity of $C$ is ______.

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