
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
In triangle $ABC$, where side lengths are $a = 6$, $b = 8$, and $c = 10$. Let $H$ be the orthocenter and $\Omega$ the circumcenter.

Determine the coordinates of the orthocenter and circumcenter expressed in the form:
\[
\overrightarrow{H} = p \overrightarrow{A} + q \overrightarrow{B} + r \overrightarrow{C}
\]
and
\[
\overrightarrow{\Omega} = u \overrightarrow{A} + v \overrightarrow{B} + w \overrightarrow{C},
\]
where $p, q, r, u, v, w$ are constants such that $p + q + r = 1$ and $u + v + w = 1$. Provide the ordered pairs $(p, q, r)$ and $(u, v, w)$.

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