
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 $a_0 = -1, b_0 = 2$, and for $n \geq 0$, define the sequences by:
\[
a_{n+1} = a_n b_n + \sqrt{a_n^2 + b_n^2}, \quad b_{n+1} = a_n b_n - \sqrt{a_n^2 + b_n^2}.
\]
Find $\frac{1}{a_{100}} + \frac{1}{b_{100}}$.

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