
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 an isosceles triangle $ABC$ with base $AB$ and equal sides $AC = BC = s$, find the locus of points $P$ in the plane such that the sum of the squares of the distances from $P$ to the vertices ($A$, $B$, and $C$) of the triangle is exactly $k\cdot s^2$ for some constant $k > 1$. Moreover, calculate the area of triangle $ABC$ if the height from $C$ to $AB$ is $h$.

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