
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 $P_1: y = x^2 + \frac{151}{150}$ and $P_2: x = y^2 + \frac{55}{5}$ be two parabolas in the Cartesian plane. Determine the equation of the common tangent line $\mathcal{L}$ to $P_1$ and $P_2$ that has a rational slope. If $\mathcal{L}$ is expressed in the form $ax + by = c$ where $a, b, c$ are positive integers and $\gcd(a, b, c) = 1$, find the value of $a + b + c$.

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