
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 $XYZ,$ $x = 13,$ $y = 15,$ and $z = 6.$ Let $I$ be the incenter and $G$ be the centroid of the triangle.
Calculate the coordinates of $I$ and $G$ given that $\overrightarrow{I} = a \overrightarrow{X} + b \overrightarrow{Y} + c \overrightarrow{Z}$ and $\overrightarrow{G} = p \overrightarrow{X} + q \overrightarrow{Y} + r \overrightarrow{Z}$, where $a, b, c, p, q, r$ are constants such that $a + b + c = 1$ and $p + q + r = 1$. Enter the ordered pairs $(a, b, c)$ and $(p, q, r)$.

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