
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 plane vectors $\overrightarrow{a}, \overrightarrow{b}, \overrightarrow{c}$ satisfy $|\overrightarrow{a}| = |\overrightarrow{b}| =1$, $\overrightarrow{a} \perp (\overrightarrow{a} - 2\overrightarrow{b})$, and $(\overrightarrow{c} - 2\overrightarrow{a}) \cdot (\overrightarrow{c} - \overrightarrow{b}) = 0$, find the sum of the maximum and minimum values of $|\overrightarrow{c}|$.

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