
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 the vectors $\overrightarrow{a}$, $\overrightarrow{b}$, and $\overrightarrow{c}$ that satisfy $|\overrightarrow{a}| = 4, |\overrightarrow{b}| = 2\sqrt{2}, \langle \overrightarrow{a}, \overrightarrow{b} \rangle = \frac{\pi}{4}$, and $(\overrightarrow{c} - \overrightarrow{a}) \cdot (\overrightarrow{c} - \overrightarrow{b}) = -1$, find the maximum value of $|\overrightarrow{c} - \overrightarrow{a}|$.

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