
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 vectors $\overrightarrow{a}=(1,-1)$, $\overrightarrow{b}=(1,1)$, $\overrightarrow{c}=(\sqrt{2}\cos \alpha, \sqrt{2}\sin \alpha) (\alpha \in \mathbb{R})$, and real numbers $m$, $n$ satisfying $m \overrightarrow{a}+n \overrightarrow{b}=2 \overrightarrow{c}$, determine the maximum value of $(m-4)^2+n^2$.

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