
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
For real numbers  $x$  and  $y$  we define  $M(x, y)$  to be the maximum of the three numbers  $xy$ ,  $(x- 1)(y - 1)$ , and  $x + y - 2xy$ . Determine the smallest possible value of  $M(x, y)$  where  $x$  and  $y$  range over all real numbers satisfying  $0 \le x, y \le 1$ .

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