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

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