
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 $DEF$, the side lengths are $d = 10$, $e = 6$, and $f = 8$. Let $J$ be the incenter of triangle $DEF$. Additionally, let the circumcenter be $O$. Given that $OJ$ is perpendicular to $DE$, find the coordinates of the incenter $J$ expressed as a linear combination of $\overrightarrow{D}$, $\overrightarrow{E}$, and $\overrightarrow{F}$, i.e., find $x$, $y$, and $z$ such that
\[
\overrightarrow{J} = x \overrightarrow{D} + y \overrightarrow{E} + z \overrightarrow{F},
\]
where $x + y + z = 1$.

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