
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 $PQR$, where $PQ = 6$, $QR = 10$, and $RP = 8$, find the incenter $I$ of the triangle using vector notation:
\[\overrightarrow{I} = x \overrightarrow{P} + y \overrightarrow{Q} + z \overrightarrow{R},\]
where $x, y, z$ are constants satisfying $x + y + z = 1$. Compute the ordered triple $(x, y, z)$.

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