
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 \( C: \frac{x^{2}}{25}+\frac{y^{2}}{9}=1 \) and a moving circle \( \Gamma: x^{2}+y^{2}=r^{2}(3<r<5) \). Let \( M \) be a point on the ellipse \( C \) and \( N \) be a point on the moving circle \( \Gamma \), such that the line segment \( MN \) is tangent to both the ellipse \( C \) and the moving circle \( \Gamma \). Find the maximum distance \( |MN| \) between points \( M \) and \( N \).

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