
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 $P_{1}: y = x^{2} + \frac{52}{25}$ and $P_{2}: x = y^{2} + \frac{81}{16}$ be two parabolas in the Cartesian plane. Let $\mathcal{L}$ be the common tangent line of $P_{1}$ and $P_{2}$ that has a rational slope. If $\mathcal{L}$ is written in the form $ax + by = c$ for positive integers $a,b,c$ where $\gcd(a,b,c) = 1$, find $a + b + c$.

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