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One Reflection Implies Another

Age 14 to 16
Challenge Level

This problem builds on from 'A Roll of Patterned Paper'

For the unit shapes I've tried so far I noticed something :

When a strip maps to itself with a mirror line somewhere across it, there always seems to be a second place where a mirror line would also map the pattern to itself.

An illustration may help :
Two vertical mirror lines
The mirror could be at position 1 or at position 2, either way the reflection will map the pattern onto itself.

Your challenge :

Can you find a design with only one mirror line across the strip?

Or perhaps, if you thought that was impossible, can you explain how you could be sure about that?

If you're ready for more, try the problem called 'Rotations Aren't Single Round Here'.