
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, 1), B(1, -1), and C($\sqrt{2} \cos\theta, \sqrt{2} \sin\theta$) where $\theta \in \mathbb{R}$, and O as the origin of the coordinate system.
1. If $|\vec{BC} - \vec{BA}| = \sqrt{2}$, find the value of $\sin 2\theta$;
2. If real numbers m and n satisfy $m \vec{OA} + n \vec{OB} = \vec{OC}$, find the maximum value of $(m-3)^2 + n^2$.

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