
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
We have  $2012$  sticks with integer length, and sum of length is  $n$ . We need to have sticks with lengths  $1,2,....,2012$ . For it we can break some sticks ( for example from stick with length  $6$  we can get  $1$  and  $4$ ).
For what minimal  $n$  it is always possible?

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