
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
Let  $\mathcal{S}$  be the set of all positive integers which are both a multiple of  $3$  and have at least one digit that is a  $1$ . For example,  $123$  is in  $\mathcal{S}$  and  $450$  is not. The probability that a randomly chosen  $3$ -digit positive integer is in  $\mathcal{S}$  can be written as  $\tfrac{m}{n}$ , where  $m$  and  $n$  are relatively prime positive integers. Find  $m+n$ .

*Proposed by GammaZero*

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