
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 \( ABC \), \( D \) lies on \( \overline{AC} \) such that \( AD:DC = 3:2 \), and \( G \) lies on \( \overline{AB} \) such that \( AG:GB = 2:3 \). \( H \) lies on \( \overline{BC} \) such that \( BH:HC = 1:2 \). Let \( Q \) be the intersection of \( \overline{DG} \) and \( \overline{AH} \). Calculate \( \overrightarrow{Q} \) using \( \overrightarrow{A} \), \( \overrightarrow{B} \), and \( \overrightarrow{C} \) and determine the coefficients \( x, y, \) and \( z \) such that \( \overrightarrow{Q} = x \overrightarrow{A} + y \overrightarrow{B} + z \overrightarrow{C}\) given \( x + y + z = 1 \).

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