
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 function $f(x) = \sin{2x} + \sin{x}\cos{x}$ ($x \in \mathbb{R}$),

1. If $x \in (0, \frac{\pi}{2})$, find the maximum value of $f(x)$;
2. In triangle $\triangle ABC$, if $A < B$ and $f(A) = f(B) = \frac{\sqrt{2}}{2}$, find the value of $\frac{\sin{C}}{\sin{A}}$.

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