
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 $\overrightarrow{m}=(1,3\cos \alpha)$ and $\overrightarrow{n}=(1,4\tan \alpha)$, where $\alpha \in (-\frac{\pi}{2}, \;\;\frac{\pi}{2})$, and their dot product $\overrightarrow{m} \cdot \overrightarrow{n} = 5$.
(I) Find the magnitude of $|\overrightarrow{m}+ \overrightarrow{n}|$;
(II) Let the angle between vectors $\overrightarrow{m}$ and $\overrightarrow{n}$ be $\beta$, find the value of $\tan(\alpha + \beta)$.

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