
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 \( x^2 + y^2 = 1 \) 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 \( C_2 \). 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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