Consider the set $L$ of binary strings of length less than or equal to $9$. For a string $w$, define $w^{+}$ to be the set $\{w, w^2, w^3, \ldots\}$ where $w^k$ represents $w$ concatenated to itself $k$ times. How many ways are there to pick an ordered pair of (not necessarily distinct) elements $x, y \in L$ such that $x^{+} \cap y^{+} \neq \varnothing$?

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