
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 \(\frac{x^{2}}{m^{2}}+\frac{y^{2}}{n^{2}}=1 (m>0, n>0)\) passing through the point \(P(3 \sqrt{3}, 1)\), find the minimum value of \(m+n\) and determine the equation of the ellipse when \(m+n\) takes its minimum value.

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