
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
Given an equilateral triangle $\triangle ABC$. Points $D$ and $E$ are taken on side $BC$ such that $BC = 3DE$. Construct an equilateral triangle $\triangle DEF$ and connect point $A$ to point $F$. Draw $DG$ parallel to $AF$ intersecting side $AB$ at point $G$, and draw $EH$ parallel to $AF$ intersecting side $AC$ at point $H$. Draw perpendiculars $GI \perp AF$ and $HJ \perp AF$. Given that the area of $\triangle BDF$ is $45$ and the area of $\triangle DEF$ is $30$, find the ratio $GI \div HJ$.

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