Pushdown automaton for = { ∈ {, }∗∣#() ≠ 2 ⋅ #()}

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I have been trying to build pushdown automaton but no clue or idea on how to start.

I got the solution but I cannot get it,

Solution I received

What I was thinking is $2$ possible variations #() < 2 ⋅ #() Or #() > 2 ⋅ #()

If someone can just explain me this solution ill be more that happy (: