
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
A sequence $(a_1, b_1)$, $(a_2, b_2)$, $(a_3, b_3)$, ..., of points in the coordinate plane satisfies
\[(a_{n+1}, b_{n+1}) = (\sqrt{3} a_n - b_n, a_n + \sqrt{3} b_n)\] for \(n = 1,2,3,\ldots\). Suppose that $(a_{50}, b_{50}) = (1, \sqrt{3})$. What is $a_1 + b_1$? Express your answer using exponential notation.

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