
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 points \((0,0), (1,2), (2,1), (2,2)\) in the plane are colored red, while the points \((1,0), (2,0), (0,1), (0,2)\) are colored blue. Four segments are drawn such that each segment connects a red point to a blue point, and each colored point is the endpoint of one segment. The smallest possible sum of the lengths of the segments can be expressed as \(a + \sqrt{b}\), where \(a\) and \(b\) are positive integers. Compute \(100a + b\).

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