
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
\[(a_{n + 1}, b_{n + 1}) = (\sqrt{3}a_n - 2b_n, 2a_n + \sqrt{3}b_n)\]
for $n = 1,2,3,\ldots$. Suppose that $(a_{50},b_{50}) = (1,3)$. What is $a_1 + b_1$?  Express your answer using exponential notation.

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