
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 equation $t^{2}+4at+3a+1=0 (a > 1)$ with two roots $\tan \alpha$ and $\tan \beta$, where $\alpha, \beta \in (-\frac{\pi}{2}, \frac{\pi}{2})$ and $x = \alpha + \beta$.
(1) Find the value of $\tan x$;
(2) Find the value of $\frac{\cos 2x}{\sqrt{2} \cos (\frac{\pi}{4} + x) \sin x}$.

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