One day, I went for a walk in the morning at $x$ minutes past 5 o'clock, where $x$ is a two-digit number. When I returned, it was $y$ minutes past 6 o'clock. I noticed the following:

1. I walked for exactly $x$ minutes.
2. $y$ was a two-digit number obtained by reversing the digits of $x$.

How many minutes did I walk?

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