
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 $DEF$, side $DE$ measures 36 units, and side $DF$ measures 45 units. Point $G$ lies on $EF$ such that $DG$ bisects $\angle EDF$. Find the ratio of the area of $\triangle DGF$ to the area of $\triangle DGE$. Provide your answer as a simplified common fraction.

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