
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
Hexagon $ABCDEF$ is equiangular with an angle of $120^\circ$ at each vertex. Square $MNOP$ is inscribed within this hexagon such that point $M$ lies on $BC$, $N$ on $DE$, and $O$ on $EF$. The side lengths of the hexagon are given as $AB=45$, and $EF=45(\sqrt{3}-1)$. Determine the side length of the square $MNOP$.
A) $45 \sqrt{2}$
B) $22.5 \sqrt{3}$
C) $45 \sqrt{3}$
D) $37.5 \sqrt{3}$
E) $40 \sqrt{2}$

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