
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 $AG:GB = 1:3,$ and $H$ lies on $\overline{BC}$ such that $BH:HC = 2:1.$ Let $Q$ be the intersection of $\overline{AG}$ and $\overline{CH}.$

Find the vector $\overrightarrow{Q}$ expressed as $u \overrightarrow{A} + v \overrightarrow{B} + w \overrightarrow{C},$ where $u,$ $v,$ and $w$ are constants such that $u + v + w = 1.$

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