
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
Mac is trying to fill 2012 barrels with apple cider. He starts with 0 energy. Every minute, he may rest, gaining 1 energy, or if he has \( n \) energy, he may expend \( k \) energy \((0 \leq k \leq n)\) to fill up to \( n(k+1) \) barrels with cider. What is the minimal number of minutes he needs to fill all the barrels?

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