Example 2 Let $X=\{1,2, \cdots, 100\}, A$ be a subset of $X$, if for any two elements $x$ 、 $y(x<y)$ in $A$, we have $y \neq 3 x$, find the maximum value of $|A|$ .
Let $A=\{3 k+1 \mid k=0,1,2, \cdots, 33\} \cup\{3 k+2 \mid k=0,1,2, \cdots, 32\} \bigcup\{9,18,36,45,63,72,81,90,99\}$, then $A$ clearly meets the conditions, at this point $|A|=76$.

On the other hand, consider 24 sets $A_{k}=\{k, 3 k\}(k=1,2,12,13, \cdots, 33)$, they contain 48 distinct numbers. Excluding these numbers, there are still 52 numbers in $X$. Each of these 52 numbers can be made into a single-element set, together with the previous 24 sets, there are a total of $52+24=76$ sets. If $|A|>76$, then $A$ must contain 2 numbers from some set $A_{k}$, where the larger number is three times the smaller number, which is a contradiction.

In summary, the maximum value of $|A|$ is 76.