
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
Consider a coordinate system where triangle $FGH$ has a right angle at $H$, $FG=5$, and $GH=12$. Triangle $FHI$ has a right angle at $F$ and $FI=15$. Points $H$ and $I$ are on opposite sides of $\overline{FG}$. The line through $I$ parallel to $\overline{FG}$ meets $\overline{GH}$ extended at $J$. Calculate the ratio $\frac{IJ}{IF}$.

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