
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
The diagonals of a convex quadrilateral form an angle of $45^{\circ}$. Perpendiculars are drawn from each vertex of the quadrilateral to the lines containing the diagonals that connect the adjacent vertices. What is the ratio of the area of the quadrilateral formed by the feet of these perpendiculars to the area of the original quadrilateral?

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