
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 $p$ and $q$ be positive integers. If $\frac{p}{q} = 0.123456789 \ldots$ (i.e., when $\frac{p}{q}$ is expressed as a decimal the first 9 digits after the decimal point are 1 to 9 in order), find the smallest value of $q$.

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