
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 \( C_2 \) have equations \( (x-1)^2 + (y-1)^2 = 4 \) and \(  (x-4)^2 + (y-1)^2 = 9 \), respectively. Find the locus of centers \( (a,b) \) of all circles externally tangent to \( C_1 \) and internally tangent to \( C_2 \). Provide your answer in the form
\[ Pa^2 + Qb^2 + Ra + Sb + T = 0, \]
where the coefficients are integers, \( P \) is positive, and \( \text{gcd}(|P|, |Q|, |R|, |S|, |T|) = 1 \).

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