
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
On the arc $AC$ of the circumcircle around an equilateral triangle $ABC$, a point $M$ is chosen such that the lengths of segments $MA$ and $MC$ are 2 and 3, respectively. The line $BM$ intersects side $AC$ at point $N$. Find the length of segment $MN$ and the side length of triangle $ABC$.

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