
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 a circle $C: x^2 + y^2 +2x - 6y + 1 = 0$, and a line $l: x + my = 3$.
(1) If $l$ is tangent to $C$, find the value of $m$;
(2) Determine if there is a value for $m$ such that $l$ intersects $C$ at points $A$ and $B$, and $\overrightarrow{OA} \cdot \overrightarrow{OB} = 0$ (where $O$ is the origin). If such an $m$ exists, find it; if not, explain why.

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