
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 the line $l: ax-2y+2=0 \ (a \in \mathbb{R})$  
(1) If it is parallel to the line $m: x+(a-3)y+1=0 \ (a \in \mathbb{R})$, find $a$;  
(2) If line $l$ always bisects the perimeter of the circle $C: (x-1)^2+y^2=2$, find $a$.

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