
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 $AB = 6,$ $BC = 8,$ and $CA = 4,$ find the coordinates of the incenter $I$ using vectors. Here, the ordered triple $(x, y, z)$ represents the vector $\overrightarrow{I} = x \overrightarrow{A} + y \overrightarrow{B} + z \overrightarrow{C}$ where $x, y,$ and $z$ are constants such that $x + y + z = 1.$

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