
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 $a_0=1,b_0=-3$, and for $n\geq 0$, let
\[
a_{n+1} = k(a_n + b_n + \sqrt{a_n^2 + b_n^2}),\quad b_{n+1} = k(a_n + b_n - \sqrt{a_n^2 + b_n^2}),
\]
where $k$ is a constant. Find $\frac{1}{a_{1000}} + \frac{1}{b_{1000}}$ if $k = \frac{1}{2}$.

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