
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
Consider two circles $C_3$ and $C_4$, where $C_3$ is given by $x^2 + y^2 = 1$ and $C_4$ given by the equation $(x - 3)^2 + y^2 = 81$. Determine the locus of 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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