
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 $\overrightarrow{a} = (4\cos\alpha, \sin\alpha)$, $\overrightarrow{b} = (\sin\beta, 4\cos\beta)$, $\overrightarrow{c} = (\cos\beta, -4\sin\beta)$ (where $\alpha, \beta \in \mathbb{R}$ and $\alpha, \beta, \alpha+\beta$ are not equal to $\frac{\pi}{2} + k\pi, k \in \mathbb{Z}$).

(Ⅰ) Find the maximum value of $|\overrightarrow{b} + \overrightarrow{c}|$;

(Ⅱ) When $\overrightarrow{a} \parallel \overrightarrow{b}$ and $\overrightarrow{a} \perp (\overrightarrow{b} - 2\overrightarrow{c})$, find the value of $\tan\alpha + \tan\beta$.

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