
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 an arbitrary positive integer  $m$ , not divisible by  $3$ , consider the permutation  $x \mapsto 3x \pmod{m}$  on the set  $\{ 1,2,\dotsc ,m-1\}$ . This permutation can be decomposed into disjointed cycles; for instance, for  $m=10$  the cycles are  $(1\mapsto 3\to 9,\mapsto 7,\mapsto 1)$ ,  $(2\mapsto 6\mapsto 8\mapsto 4\mapsto 2)$  and  $(5\mapsto 5)$ . For which integers  $m$  is the number of cycles odd?

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