[Solved] Very long equation manipulation with arrows

Stefano Asks: Very long equation manipulation with arrows
I was trying to fit this equation manipulation in a page as shown in the MWE:

Code:
documentclass{article}
usepackage[utf8]{inputenc}
usepackage{geometry} geometry{a4paper,top=2.5cm,bottom=2.5cm,left=2cm,right=2cm,heightrounded,bindingoffset=5mm}

usepackage{amsmath,cancel,witharrows}

begin{document}

noindent
some words
begin{DispWithArrows*}
    P(t|t)&=Eleft[tilde{e}_x(t)tilde{e}_x^T(t)right]\
    &=Eleft[big((I-K(t)C)e_x(t)-K(t)v(t)big)big((I-K(t)C)e_x(t)-K(t)v(t)big)^Tright] Arrow[i]{$e_x$ is uncorrelated with $v(t)$\collect $E[e_x(t)e_x^T(t)]$ and $E[v(t)v^T(t)]$}\
    &=big(I-K(t)Cbig)E[e_x(t)e_x^T(t)]big(I-K(t)Cbig)^T+K(t)E[v(t)v^T(t)]K(t)\
    &=big(I-K(t)Cbig)P(t|t-1)big(I-K(t)Cbig)^T+K(t)R_vK^T(t)\
    &begin{split}
        =P(t|t-1)-P(t|t-1)C^TK^T(t)-K(t)CP(t|t-1)+\
        +K(t)CP(t|t-1)C^TK^T(t)+K(t)R_vK^T(t)
    end{split}
    Arrow[i]{in the 4th and 5th terms,\factor out $K(t)$}\
    &begin{split}
        =P(t|t-1)-P(t|t-1)C^TK^T(t)-K(t)CP(t|t-1)+\
        +K(t)big(CP(t|t-1)C^T+R_vbig)K^T(t)
    end{split}    
    Arrow[i]{eqref{eqn_kalman_filter_gain}}\
    &begin{split}=P(t|t-1)-P(t|t-1)C^TK^T(t)-K(t)CP(t|t-1)+\
    +underbrace{P(t|t-1)C^Tbig(CP(t|t-1)C^T+R_vbig)^{-1}}_{=K(t)}big(CP(t|t-1)C^T+R_vbig)K^T(t)end{split}\
    &=P(t|t-1)cancel{-P(t|t-1)C^T K^T(t)}-K(t)CP(t|t-1)cancel{+P(t|t-1)C^TK^T(t)}\
    &=P(t|t-1)-K(t)CP(t|t-1)Arrow[i]{factor out $P(t|t-1)$}\
    P(t|t)&=big(I-K(t)Cbig)P(t|t-1)
end{DispWithArrows*}
some other words

end{document}

Problem 1: The text displayed with the package witharrows and with the command Arrow[i]{...} falls off the right edge of the sheet, while the equation doesn’t start immediately at left but in the center-left of the sheet, so it doesn’t exploit all the available space.

Problem 2: I tried to use the split environment but it doesn’t work very well because some equations don’t start aligned with the others (I think it’s because I don’t use it properly) and because its efforts are useless due to problem 1.

If my description is not clear, the output of the MWE will be clearer:


if my description is not clear, the output of the MWE will be clearer

How can I solve these problems?

Thanks in advance!

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