
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
Suppose \( a_{0}=a \), and \( a_{n+1}=2a_{n}-n^{2} \) for \( n=0,1,2,\cdots \). If all terms of the sequence \( \{a_{n}\} \) are positive integers, what is the minimum value of \( \{a_{n}\} \)?

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