
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 $ABC,$ $G$ lies on $\overline{AB}$ such that $AG:GB = 3:2,$ and $H$ lies on $\overline{BC}$ such that $BH:HC = 1:3.$ Let $Q$ be the intersection of $\overline{AG}$ and $\overline{CH}.$ Determine the vector $\overrightarrow{Q}$ in terms of $\overrightarrow{A}, \overrightarrow{B},$ and $\overrightarrow{C}$ such that $\overrightarrow{Q} = x \overrightarrow{A} + y \overrightarrow{B} + z \overrightarrow{C}$, where $x, y,$ and $z$ are constants satisfying $x + y + z = 1.$

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