
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
Let $x$ and $y$ be non-zero real numbers such that $\frac{x \sin \frac{\pi}{5} + y \cos \frac{\pi}{5}}{x \cos \frac{\pi}{5} - y \sin \frac{\pi}{5}} = \tan \frac{9\pi}{20}$.
(1) Find the value of $\frac{y}{x}$;
(2) In triangle $ABC$, if $\tan C=\frac{y}{x}$, find the maximum value of $\sin 2A + 2 \cos B$.

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