
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 $H$ be the orthocenter of triangle $DEF$ with side lengths $d$, $e$, and $f$, and circumradius $S$. For all points $Q$ on the circumcircle of triangle $DEF$,
\[ QD^2 + QE^2 + QF^2 - QH^2 \]
is a constant. Express this constant in terms of the side lengths $d$, $e$, $f$, and circumradius $S$ of triangle $DEF$.

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