Recall that $[x]$ denotes the largest integer not exceeding $x$ for real $x$. For integers $a, b$ in the interval $1 \leq a < b \leq 100$, find the number of ordered pairs $(a, b)$ satisfying the following equation:

$$\left[a+\frac{b}{a}\right]=\left[b+\frac{a}{b}\right]$$

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