
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
Two circles $\mathcal{C}_{1}$ and $\mathcal{C}_{2}$ intersect at $(2, 3)$ and another unknown point. The product of the radii of the circles is $50$. The y-axis and the line $y = nx$ (where $n > 0$) are tangent to both circles. Find $n$ in the simplest form $p\sqrt{q}/r$, where $p$, $q$, and $r$ are positive integers with $q$ square-free and $p$ and $r$ relatively prime. Determine $p + q + r$.

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