
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 the left and right foci $F_{1}$, $F_{2}$, the circle with diameter $F_{1}F_{2}$ intersects $C$ at points $A$ and $B$ in the first and third quadrants, respectively. If $|AF_{1}| \leq 4|BF_{1}|$, then the range of eccentricity of $C$ is ______.

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