
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 $\mathcal{P}$ be the parabola defined by the equation $y = x^2$. A circle $\mathcal{C}$ intersects $\mathcal{P}$ at four distinct points. Given three of these points as $(-3,9)$, $(1,1)$, and $(4,16)$, find the sum of the distances from the focus of $\mathcal{P}$ to all four intersection points.

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