
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 points A(1, 0), B(0, 1), and C(2sinθ, cosθ).
(1) If $|\overrightarrow{AC}| = |\overrightarrow{BC}|$, find the value of $\frac{\sin\theta + 2\cos\theta}{\sin\theta - \cos\theta}$;
(2) If $(\overrightarrow{OA} + 2\overrightarrow{OB}) \cdot \overrightarrow{OC} = 1$, where O is the origin, find the value of $\sin\theta \cdot \cos\theta$.

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