
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
The point F(c,0) is the right focus of the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1 (a > 0, b > 0)$. Point P is a point on the left branch of the hyperbola. The line segment PF is tangent to the circle $x^{2}+y^{2}=\frac{b^{2}}{4}$ at point Q, and $\overrightarrow{PQ}=\frac{1}{2}\overrightarrow{PF}$. Determine the eccentricity of the hyperbola.

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