
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
Let the vectors $\overrightarrow{a} = (4\cos\alpha, \sin\alpha)$, $\overrightarrow{b} = (\sin\beta, 4\cos\beta)$, $\overrightarrow{c} = (\cos\beta, -4\sin\beta)$.
(1) If $\overrightarrow{a}$ is perpendicular to $\overrightarrow{b} - 2\overrightarrow{c}$, find the value of $\tan(\alpha + \beta)$;
(2) Find the maximum value of $|\overrightarrow{b} + \overrightarrow{c}|$.

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