
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
On the sides \( AB \) and \( BC \) of rectangle \( ABCD \), points \( F \) and \( G \) are chosen, respectively. From point \( F \), a perpendicular \( FK \) is dropped to side \( CD \). From point \( G \), a perpendicular \( GH \) is dropped to side \( AD \). The intersection of \( FK \) and \( GH \) is denoted by \( E \). Find the area of triangle \( DFG \), given that the areas of rectangles \( ABCD \) and \( HEKD \) are 20 and 8, respectively.

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