
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
Given a right-angled triangle with vertices placed at $(0, 0)$, $(a, 0)$, and $(0, b)$ (where $a$ and $b$ are positive numbers), find the set of all points $P$ in the plane such that the sum of the squares of the distances from $P$ to each of the triangle's vertices is less than $k^2$.
**A)** Circle with radius $\sqrt{\frac{k^2 - a^2 - b^2}{3}}$.
**B)** Circle with radius $\sqrt{\frac{k^2 - \frac{2a^2}{3} - \frac{2b^2}{3}}{3}}$.
**C)** Circle with radius $\sqrt{\frac{k^2 - 2a^2 - 2b^2}{3}}$.
**D)** No set exists.

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