
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
Let \(\left(a_{n}\right)\) and \(\left(b_{n}\right)\) be sequences defined by \(a_{0}=b_{0}=0\) and \(a_{n}=a_{n-1}^{2}+3\) and \(b_{n}=b_{n-1}^{2}+2^{n}\).

Compare the numbers \(a_{2003}\) and \(b_{2003}\).

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