
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
The circle \( C: x^{2}+(y-2)^{2}=16 \) is symmetric about the line \( a x + b y - 12 = 0 \). A moving point \( S \) lies on the line \( y + b = 0 \). From point \( S \), two tangents \( S A \) and \( S B \) are drawn to the circle \( C \), touching the circle at points \( A \) and \( B \) respectively. The line \( AB \) always passes through a fixed point. Find the coordinates of this fixed point.

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