
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 with radius 2 whose center is on the positive x-axis, the line $3x-4y+4=0$ is tangent to circle C.
1. Find the equation of circle C.
2. A line l passing through point Q (0, -3) with slope k intersects circle C at two distinct points A $(x_1, y_1)$ and B $(x_2, y_2)$,
   a. When k=3, calculate the value of $x_1 \cdot x_2 + y_1 \cdot y_2$;
   b. When $x_1 \cdot x_2 + y_1 \cdot y_2 = 8$, find the equation of line l.

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