
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 $ABC,$ $a = 8,$ $b = 10,$ and $c = 6.$ Let $I$ be the incenter. Compute the position vector $\overrightarrow{I}$ in terms of the vectors $\overrightarrow{A}$, $\overrightarrow{B}$, and $\overrightarrow{C}$, where $x, y, z$ are constants such that $x + y + z = 1$.

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