
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 $\overrightarrow{a} = (\cos \alpha, \sin \alpha)$, $\overrightarrow{b} = (\cos \beta, \sin \beta)$, and $\overrightarrow{c} = (-1, 0)$:  
(1) Find the maximum length of the vector $\overrightarrow{b} + \overrightarrow{c}$;  
(2) If $\alpha = \frac{\pi}{4}$ and $\overrightarrow{a}$ is perpendicular to $\overrightarrow{b} + \overrightarrow{c}$, find the value of $\cos \beta$.

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