
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
1. In the diagram, square \( EFGH \) is centered inside square \( ABCD \) and the region between the two squares is shaded.
(a) Determine the area of the shaded region if the length of \( AB \) is \( 10 \) cm and the length of \( EF \) is \( 8 \) cm.
(b) Determine the length of \( EF \) if the length of \( AB \) is \( 13 \) cm and the area of the shaded region is \( 120 \) cm\(^2\).
(c) Determine the length of \( EF \) if the length of \( AB \) is \( 18 \) cm and the area of the shaded region is \(\frac{3}{4}\) of the area of \( ABCD \).
(d) Suppose that the area of the shaded region is \( 64 \% \) of the area of \( ABCD \). If the length of \( AB \) is \( a \) cm and the length of \( EF \) is \( b \) cm, determine the value of \(\frac{a}{b}\).

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