
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 \(D\), \(E\), and \(F\) be the perpendicular distances from point \(P\) on the circumcircle of triangle \(ABC\) to the sides \(BC\), \(CA\), and \(AB\) respectively. Given that \(a\), \(b\), and \(c\) are the lengths of the sides opposite to vertices \(A\), \(B\), and \(C\) respectively, and \(R\) is the circumradius of \(ABC\), find the constant value of
\[
a^2D^2 + b^2E^2 + c^2F^2
\]
in terms of \(a\), \(b\), \(c\), and \(R\).

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