
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 $XYZ,$ point $G$ is on segment $\overline{XZ}$ such that the ratio $XG:GZ = 3:2,$ and point $H$ is on segment $\overline{XY}$ such that the ratio $XH:HY = 1:3.$ Let point $Q$ be the intersection of $\overline{ZG}$ and $\overline{YH}$.
Using vector notation:
\[ \overrightarrow{Q} = a \overrightarrow{X} + b \overrightarrow{Y} + c \overrightarrow{Z}, \]
where $a, b,$ and $c$ are constants that satisfy $a + b + c = 1.$ Determine the ordered triple $(a, b, c)$.

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