
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
Let \( a \) and \( b \) be positive real numbers with \( a \ge b \). Determine the maximum possible value of \( \frac{a}{b} \) for which the following conditions are met:
1. The points \( (0,0) \), \( (a, y) \), \( (\frac{a}{2}, b) \), and \( (x, 0) \) form a kite, where \( 0 \le x < a \) and \( 0 \le y < b \).
2. The diagonals of the kite are \( a^2 + y^2 \) and \( b^2 + x^2 \), and they are orthogonal.

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