
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 the plane, fixed points $A$, $B$, $C$, $D$ satisfy $|\overrightarrow{DA}| = |\overrightarrow{DB}| = |\overrightarrow{DC}|$, $\overrightarrow{DA} \cdot \overrightarrow{DB} = \overrightarrow{DB} \cdot \overrightarrow{DC} = \overrightarrow{DC} \cdot \overrightarrow{DA} = -2$. For moving points $P$, $M$ satisfying $|\overrightarrow{AP}| = 1$, $\overrightarrow{PM} = \overrightarrow{MC}$, the maximum value of $|\overrightarrow{BM}|^2$ is \_\_\_\_.

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