
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 \( \triangle ABC \), 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}
user