
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
(1) If $\cos (\frac{\pi}{4}+x) = \frac{3}{5}$, and $\frac{17}{12}\pi < x < \frac{7}{4}\pi$, find the value of $\frac{\sin 2x + 2\sin^2 x}{1 - \tan x}$.

(2) Given the function $f(x) = 2\sqrt{3}\sin x\cos x + 2\cos^2 x - 1 (x \in \mathbb{R})$, if $f(x_0) = \frac{6}{5}$, and $x_0 \in [\frac{\pi}{4}, \frac{\pi}{2}]$, find the value of $\cos 2x_0$.

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