
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
Let \( P_1: y = x^2 + \frac{169}{100} \) and \( P_2: x = y^2 + \frac{49}{4} \) be two parabolas in the Cartesian plane. Find a common tangent line \( \mathcal{L} \) of \( P_1 \) and \( P_2 \) that has a rational slope. If \( \mathcal{L} \) can be written in the form \( ax + by = c \) for positive integers \( a, b, c \) where \( \gcd(a, b, c) = 1 \), determine \( a + b + c \).

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