
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
In this sentence, the number of numbers appearing 1 time is $a_{1}$, the number of numbers appearing 2 times is $a_{2}$, $\ldots$, the number of numbers appearing 2013 times is $a_{2013}$. Determine the numbers $a_{1}, a_{2}, \ldots, a_{2013}$ such that the statement is true. In how many ways can this be done?

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