
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 \( a, b \) be positive reals with \( a > b > \frac{1}{2}a \). Place two squares of side lengths \( a, b \) next to each other, such that the larger square has its lower left corner at \( (0,0) \) and the smaller square has its lower left corner at \( (a, 0) \). Draw the line passing through \( (0, a) \) and \( (a+b, 0) \). The region in the two squares lying above the line has an area of 2013. If \( (a, b) \) is the unique pair maximizing \( a+b \), compute \( \frac{a}{b} \).

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