
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
Define the sequence \(\{a_{n}\}\) by \(a_0 = a\), \(a_{n+1} = 2a_n - n^2\) (for \(n \geq 0\)). Determine all real numbers \(a\) such that \(a_n \geq 0\) for any non-negative real number \(n\).

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