
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
Two circles \(C_3\) and \(C_4\) are defined by the equations \(x^2 + y^2 = 4\) and \((x - 3)^2 + y^2 = 25\), respectively. Find the locus of the centers \((a,b)\) of all circles that are externally tangent to \(C_3\) and internally tangent to \(C_4\). Express your answer in the form
\[
Pa^2 + Qb^2 + Ra + Sb + T = 0,
\]
where \(P\), \(Q\), \(R\), \(S\), and \(T\) are integers, \(P\) is positive, and \(\gcd(|P|, |Q|, |R|, |S|, |T|) = 1\).

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