On a clock, there are two instants between $12$ noon and $1\,\mathrm{PM}$ when the hour hand and the minute hand are at right angles. The difference in minutes between these two instants is written as $a + \dfrac{b}{c}$, where $a, b, c$ are positive integers, with $b < c$ and $\dfrac{b}{c}$ in reduced form. What is the value of $a+b+c$?

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