
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
The circles $C_1$ and $C_2$ are defined by the equations $x^2 + y^2 = 4$ and $(x - 1)^2 + y^2 = 9$, respectively. Find the locus of the centers $(a,b)$ of all circles externally tangent to $C_1$ and internally tangent to $C_2.$ Enter 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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