
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
Two unit squares \( S_{1} \) and \( S_{2} \) have horizontal and vertical sides. Let \( x \) be the minimum distance between a point in \( S_{1} \) and a point in \( S_{2} \), and let \( y \) be the maximum distance between a point in \( S_{1} \) and a point in \( S_{2} \). Given that \( x = 5 \), the difference between the maximum and minimum possible values for \( y \) can be written as \( a + b \sqrt{c} \), where \( a, b \), and \( c \) are integers and \( c \) is positive and square-free. Find \( 100a + 10b + c \).

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