Help identifying the proper length in relativity questions?

A stationary muon has a half life of 2.2E-6s. Calculate the distance, in a lab's frame of reference, travelled by the muon in one half life given that its speed is 0.996c.

Now i know how to solve this using the time dilation equation but i need to know how to do this using the length contraction equation.

My book defines the proper length as the length measured by an observer who is stationary with respect to the event.

From my understanding, that means that the the distance travelled by the muon, in its frame of reference, is the proper length because if you're travelling at the same speed as the muon alongside it, you'd be stationary relative to it.

But it turns out that, to get the right answer, the proper length is in fact the distance measured in the lab frame of reference.

So this is how the calculation should be done:

l= 0.996x3E8x2.2E6= 657m

So, proper length, l0 =657( 1 - 0.996^2)^(-1/2)= 7360m

I am so confused! Why is the distance measured in the lab frame of reference considered the proper length? Someone please explain to me how i can figure out which length is the proper length!

Thanks so much in advance!