
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 \( E: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 \) with \( a > b > 0 \), points \( A \) and \( B \) are the left and top vertices of ellipse \( E \), respectively. The line \( AB \) is tangent to the circle \( C: x^{2}+y^{2}=\frac{c^{2}}{4} \) (where \( c \) is the semi-focal length of the ellipse). Let \( P \) be a moving point on the line \( AB \). From point \( P \), two tangents are drawn to the circle \( C \), touching the circle at points \( M \) and \( N \), respectively. If there exists a point \( P \) such that triangle \( \triangle PMN \) is an equilateral triangle, find the range of the eccentricity \( e \) of the ellipse.

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