How to express the interconnection of LTI systems as the impulse responses?

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For the interconnection of the LTI system as the diagram shows, how can we express and simplify this interconnection to one LTI system using each impulse response? interconnection of the LTI

I know that you are supposed to convolve the impulse responses when you go through two LTI in series, but how should we treat this complex feedback system with the following properties in hand? $h1[n] = h2'[n]$

$h2[n] = h4'[n]$

$h5[n] = h3'[n]$

Any help is appreciated and thanks for your time!