
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 $f(x) = 2\sin^2x + \cos\left(\frac{\pi}{3} - 2x\right)$.  
(1) Find the decreasing interval of $f(x)$ on $[0, \pi]$.  
(2) Let $\triangle ABC$ have internal angles $A$, $B$, $C$ opposite sides $a$, $b$, $c$ respectively. If $f(A) = 2$, and the vector $\overrightarrow{m} = (1, 2)$ is collinear with the vector $\overrightarrow{n} = (\sin B, \sin C)$, find the value of $\frac{a}{b}$.

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