
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 that point $P$ is any point on the ellipse $C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a>b>0)$ other than the upper and lower vertices, and $O$ is the origin, a tangent line is drawn through point $P$ to the circle $O: x^{2}+y^{2}=b^{2}$ with tangents at points $M$ and $N$. If there exists a point $P$ such that $\angle MPN=90^{\circ}$, then the minimum value of the eccentricity of the ellipse is ______.

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