
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 $Q_{1}: y = x^2 + 2$ and $Q_{2}: x = y^2 + 8$ be two parabolas in the Cartesian plane. Let $\mathcal{M}$ be the common tangent line of $Q_{1}$ and $Q_{2}$ that has a nonzero integer slope. Assume $\mathcal{M}$ is written in the form $dx + ey = f$ for integers $d, e, f$. Find $d+e+f$ if $d, e,$ and $f$ are coprime.

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