
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
In rhombus  $NIMO$ ,  $MN = 150\sqrt{3}$  and  $\measuredangle MON = 60^{\circ}$ .  Denote by  $S$  the locus of points  $P$  in the interior of  $NIMO$  such that  $\angle MPO \cong \angle NPO$ .  Find the greatest integer not exceeding the perimeter of  $S$ .

*Proposed by Evan Chen*

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