
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 $\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1\left(a \gt b \gt 0\right)$, with foci $F_{1}(-c,0)$ and $F_{2}(c,0)\left(c \gt 0\right)$. If a line passing through $F_{1}$ is tangent to the circle $(x-\frac{1}{2}c)^{2}+y^{2}=c^{2}$ and intersects the ellipse in the first quadrant at point $P$, and $PF_{2}\bot x$-axis, then the slope of the line is ____ and the eccentricity of the ellipse is ____.

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