
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
Find the locus of points $M$ on the plane such that the difference in the squares of the distances from $M$ to two given points $A$ and $B$ on this plane is constant and equal to $a$. For which values of $a$ does the problem have solutions?

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