
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 $m \overrightarrow {a}+n \overrightarrow {b}= \overrightarrow {c}$, then the maximum value of $(m-3)^2+n^2$ is \_\_\_\_\_\_.

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