
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
Given that \( P \) is a real polynomial of degree at most 2012 such that \( P(n) = 2^n \) for \( n = 1, 2, \ldots, 2012 \), what choice(s) of \( P(0) \) produce the minimal possible value of \( P(0)^2 + P(2013)^2 \)?

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