
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 a triangle $ABC$, $E$ lies on $\overline{AC}$ and $G$ lies on $\overline{AB}$. Let $\overline{BE}$ intersect $\overline{AG}$ at $Q$. If the ratio $BQ:QE = 2:1$ and $GQ:QA = 3:4$, find $\frac{BG}{GA}$.

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