A positive integer is called *Tiranian* if it can be written as $x^2 + 6xy + y^2$, where $x$ and $y$ are (not necessarily distinct) positive integers. The integer $36^{2023}$ is written as the sum of $k$ Tiranian integers. What is the smallest possible value of $k$?

Let's think step by step and output the final answer within \boxed{}.
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