
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 a function $f(x) = 2\sin x \cos x$, with $x \in \mathbb{R}$.  
(1) If $f\left(\frac{\alpha}{2}\right) = \frac{3}{5}$, with $\alpha \in \left( \frac{\pi}{2}, \pi \right)$, find the value of $\cos\left(\alpha - \frac{\pi}{3}\right)$.  
(2) Find the intervals where $f(x)$ is decreasing.  
(3) Determine the equation of the tangent line to the curve $y = f(x)$ at the origin $O$.

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