
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: $f(x) = a\sin{2x} + \cos{2x}$ and $f\left( \frac{\pi}{3} \right) = \frac{\sqrt{3}-1}{2}$.
(1) Find the value of $a$ and the maximum value of $f(x)$;
(2) Find the intervals where $f(x)$ is monotonically decreasing.

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