
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 subset  $M$  of  $\{1, 2, . . . , 2006\}$  has the property that for any three elements  $x, y, z$  of  $M$  with  $x < y < z$ ,  $x+ y$  does not divide  $z$ . Determine the largest possible size of  $M$ .

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