
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
Let  $ABCD$  be a square and  $a$  be the length of his edges. The segments  $AE$  and  $CF$  are perpendicular on the square's plane in the same half-space and they have the length  $AE=a$ ,  $CF=b$  where  $a<b<a\sqrt 3$ . If  $K$  denoted the set of the interior points of the square  $ABCD$  determine  $\min_{M\in K} \left( \max ( EM, FM ) \right) $  and  $\max_{M\in K} \left( \min (EM,FM) \right)$ . 

*Octavian Stanasila*

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