
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 \(a_0 = -4\), \(b_0 = 2\), and for \(n \geq 0\), define the sequences \(a_{n+1}\) and \(b_{n+1}\) as follows:
\[
\begin{align*}
a_{n+1} &= a_n + b_n + \sqrt{a_n^2 + b_n^2},\\
b_{n+1} &= a_n + b_n - \sqrt{a_n^2 + b_n^2}.
\end{align*}
\]
Find \(\frac{1}{a_{1011}} + \frac{1}{b_{1011}}.\)

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