
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 $\overrightarrow{a}=(4, 5\cos\alpha)$ and $\overrightarrow{b}=(3, -4\tan\alpha)$ with $\alpha \in (0, \frac{\pi}{2})$ and $\overrightarrow{a} \perp \overrightarrow{b}$, calculate:  
(1) $|\overrightarrow{a}+\overrightarrow{b}|$;  
(2) The value of $\cos(\alpha+ \frac{\pi}{4})$.

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