
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 an equilateral triangle $ABC$ with side length $2\sqrt{3}$, and a moving point $P$ in the plane of $ABC$ such that $|\overrightarrow{AP}|=1$, and $\overrightarrow{PM}=\overrightarrow{MC}$, then the maximum value of $|\overrightarrow{BM}|^{2}$ is \_\_\_\_.

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