
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 $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1({a>b>c>0})$ with left and right foci $F_{1}$ and $F_{2}$ respectively. If a circle with center at $F_{2}$ and radius $b-c$ is drawn, and a tangent line is drawn from a point $P$ on the ellipse to this circle. Let $T$ be the point of tangency, and the minimum value of $|PT|$ is $\frac{{\sqrt{3}}}{2}({a-c})$. Find the range of eccentricity $e$ of the ellipse.

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