
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 its center C on the positive x-axis and a radius of 5, the chord intercepted by the line $x-y+3=0$ has a length of $2\sqrt{17}$.
(1) Find the equation of circle C;
(2) Suppose the line $ax-y+5=0$ intersects circle C at points A and B, find the range of the real number $a$;
(3) Under the condition of (2), is there a real number $a$ such that points A and B are symmetric about the line $l$ passing through point P(-2, 4)? If it exists, find the value of the real number $a$; if not, please explain why.

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