
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
When $12^3$ is written out in base 10, the sum of its digits is $9=3^2$. What is the largest base $b$ such that the base-$b$ digits of $12^3$ do not add up to $3^2$? (Note: here $12^3$ in base $b$ means that the base-$b$ number $12$ is raised to the power of $3$.)

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