
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 with a radius of 5, its center lies on the x-axis with an integer horizontal coordinate and is tangent to the line 4x + 3y - 29 = 0.

(1) Find the equation of the circle;

(2) If the line ax - y + 5 = 0 (a ≠ 0) intersects the circle at points A and B, does there exist a real number a such that the line l passing through point P(-2, 4) is perpendicularly bisecting chord AB? If such a real number a exists, find its value; otherwise, explain the reason.

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