
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

A right triangle \(ABC (\angle A = 90^\circ)\) and two squares \(BEFC\) and \(AMNC\) are positioned such that points \(E\) and \(A\) lie on opposite sides of line \(BC\), and points \(M\) and \(B\) lie on the same side of line \(AC\). Find the distance between the centers of the squares, given \(AB = a\).

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