
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 arithmetic mean, \( A \), of any two positive numbers \( x \) and \( y \) is defined to be \( A = \frac{1}{2}(x + y) \) and their geometric mean, \( G \), is defined to be \( G = \sqrt{xy} \). For two particular values \( x \) and \( y \), with \( x > y \), the ratio \( A : G = 5 : 4 \). For these values of \( x \) and \( y \), what is the ratio \( x : y \)?  
A) \( 5 : 4 \)  
B) \( 2 : 1 \)  
C) \( 5 : 2 \)  
D) \( 7 : 2 \)  
E) \( 4 : 1 \)

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