
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 a circle $C$ with equation $(x-m)^{2}+(y-n)^{2}=9$ whose center is located in the first quadrant, and a line $l$ with equation $x+2y+2=0$. If the length of the chord formed by the intersection of line $l$ and circle $C$ is $4$, determine the minimum value of $\frac{m+2n}{mn}$.

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