
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
A number $N$ is chosen from the interval $1$ to $1500$. Calculate the probability that the remainder when $N^{12}$ is divided by $3$ is $1$.

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