
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 $f(x) = 2\sin(180^\circ - x) + \cos(-x) - \sin(450^\circ - x) + \cos(90^\circ + x)$.

(1) If $f(\alpha) = \frac{2}{3}$ and $\alpha \in (0^\circ, 180^\circ)$, find $\tan\alpha$;  
(2) If $f(\alpha) = 2\sin\alpha - \cos\alpha + \frac{3}{4}$, find the value of $\sin\alpha \cdot \cos\alpha$.

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