
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
Determine the number of positive integers  $a \le 250$  for which the set  $\{a+1, a+2, \dots, a+1000\}$  contains $\bullet$  Exactly  $333$  multiples of  $3$ , $\bullet$  Exactly  $142$  multiples of  $7$ , and $\bullet$  Exactly  $91$  multiples of  $11$ .

*Based on a proposal by Rajiv Movva*

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