
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) Let $\alpha= \frac{\pi}{4}$, and $\overrightarrow{a} \perp (\overrightarrow{b} + \overrightarrow{c})$, find the value of $\cos \beta$.

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