
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 value of the magnitude of vector $\overrightarrow{b} + \overrightarrow{c}$;
(2) Let $\alpha = \frac{\pi}{3}$ and $\overrightarrow{a} \cdot (\overrightarrow{b} + \overrightarrow{c}) = \frac{1}{2}$, find the value of $\sin\beta.$

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