
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 $$\overrightarrow {a}=(4\cos\alpha, \sin\alpha)$$, $$\overrightarrow {b}=(\sin\beta, 4\cos\beta)$$, and $$\overrightarrow {c}=(\cos\beta, -4\sin\beta)$$.

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

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