
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 the square \(ABCD\), what is the set of points \(M\) in the plane of the square for which the sum of the areas of the triangles \(MAC\) and \(MBD\) is equal to the area of the square?

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