
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
There exist positive integers  $N, M$  such that  $N$ 's remainders modulo the four integers  $6, 36,$   $216,$  and  $M$  form an increasing nonzero geometric sequence in that order. Find the smallest possible value of  $M$ .

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