
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 two base-10 positive integers  $a$  and  $b$ , we say  $a \sim b$  if we can rearrange the digits of  $a$  in some way to obtain  $b$ , where the leading digit of both  $a$  and  $b$  is nonzero. For instance,  $463 \sim 463$  and  $634 \sim 463$ . Find the number of  $11$ -digit positive integers  $K$  such that  $K$  is divisible by  $2$ ,  $3$ , and  $5$ , and there is some positive integer  $K'$  such that  $K' \sim K$  and  $K'$  is divisible by  $7$ ,  $11$ ,  $13$ ,  $17$ ,  $101$ , and  $9901$ .

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