
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 the vectors $\overrightarrow{a}=(4\cos α,\sin α)$, $\overrightarrow{b}=(\sin β,4\cos β)$, $\overrightarrow{c}=(\cos β,-4\sin β)$:
1. If $\overrightarrow{a}$ is perpendicular to $\overrightarrow{b}-2\overrightarrow{c}$, find the value of $\tan (α+β)$;
2. If $β∈(-\frac{π}{12}, \frac{5π}{12}]$, find the range of values for $| \overrightarrow{b}+ \overrightarrow{c}|$.

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