
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 $ABC,$ $F$ lies on $\overline{BC}$ extended past $C$ such that $BF:FC = 4:1,$ and $G$ lies on $\overline{AB}$ extended past $B$ such that $AG:GB = 2:1.$ Let $Q$ be the intersection of lines $CG$ and $AF.$

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