
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
Given two squares with sides parallel to each other, determine the set of points \( M \) such that for any point \( P \) in the first square, there exists a point \( Q \) in the second square such that the triangle \( MPQ \) is equilateral. Let the side lengths of the first and second squares be \( a \) and \( b \) respectively. For what relationship between \( a \) and \( b \) is the set of points non-empty?

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