
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 \(C_1\) and \(C_2\) 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 externally tangent to \(C_1\) and internally tangent to \(C_2\). Express your answer in the form
\[ Pa^2 + Qb^2 + Ra + Sb + T = 0, \]
where all the coefficients 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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