
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
Circles $\mathcal{C}_{1}$ and $\mathcal{C}_{2}$ intersect at two points, one of which is $(8,4)$. The product of the radii is $77$. The x-axis, the line $y = 2x + 3$, and the vertical line $x = p$ (where $p > 0$) are tangent to one of the circles. Determine the value of $p$.

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