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

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