
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 maximum number of natural numbers  $x_1,x_2, ... , x_m$   satisfying the conditions:
a) No  $x_i - x_j , 1 \le i < j \le m$  is divisible by  $11$ ,  and
b) The sum  $x_2x_3 ...x_m + x_1x_3  ... x_m + \cdot \cdot \cdot + x_1x_2... x_{m-1}$  is divisible by  $11$ .

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