
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 $A(3,0)$, $B(0,3)$, and $C(\cos\alpha,\sin\alpha)$, with $O$ being the origin.
(1) If $\overrightarrow{OC} \parallel \overrightarrow{AB}$, find the value of $\tan\alpha$;
(2) If $\overrightarrow{AC} \perp \overrightarrow{BC}$, find the value of $\sin 2\alpha$;
(3) If $|\overrightarrow{OA} + \overrightarrow{OC}| = \sqrt{13}$ and $\alpha \in (0,\pi)$, find the angle between $\overrightarrow{OB}$ and $\overrightarrow{OC}$.

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