
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
A circle with radius \(5\) has its center on the \(x\)-axis, and the \(x\)-coordinate of the center is an integer. The circle is tangent to the line \(4x+3y-29=0\).
(Ⅰ) Find the equation of the circle;
(Ⅱ) Let the line \(ax-y+5=0\) (\(a > 0\)) intersect the circle at points \(A\) and \(B\), find the range of values for the real number \(a\);
(Ⅲ) Under the condition of (Ⅱ), determine if there exists a real number \(a\) such that the perpendicular bisector line \(l\) of chord \(AB\) passes through point \(P(-2,4)\), and if so, find the value of \(a\); if not, explain why.

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