How to align contents of table cell to bottom?

I’ve looked at a few answers about aligning the contents of table cells vertically. In the MWE I provide, there are 4 tables within the columns of another table. The tables are of different sizes and I’d like them all aligned along the bottom of their cell.

So the table in the first column should be aligned to the bottom of the outer table.

documentclass{article}
usepackage{tikz}

newcommand{tikzmark}[2][-3pt]{tikz[remember picture, overlay, baseline=-0.5ex]node[#1](#2){};}

tikzset{brace/.style={decorate, decoration={brace}},
 brace mirrored/.style={decorate, decoration={brace,mirror}},
}

newcounter{brace}
setcounter{brace}{0}
newcommand{drawbrace}[3][brace]{%
 refstepcounter{brace}
 tikz[remember picture, overlay]draw[#1] (#2.center)--(#3.center)node[pos=0.5, name=brace-thebrace]{};
}

newcounter{arrow}
setcounter{arrow}{0}
newcommand{drawcurvedarrow}[3][]{%
 refstepcounter{arrow}
 tikz[remember picture, overlay]draw (#2.center)edge[#1]node[coordinate,pos=0.5, name=arrow-thearrow]{}(#3.center);
}

newcommand{annote}[3][]{%
 tikz[remember picture, overlay]node[#1] at (#2) {#3};
}

begin{document}
begin{table}
centering
begin{tabular}{|c|c|c|c|}
hline
 begin{tabular}{ | l |}
    $3 times fact(2)$tikzmark[xshift=0.62em]{d}  hline
    $4 times fact(3)$tikzmark[xshift=0.6em]{f} hline
 end{tabular}
 &  
 begin{tabular}{ | l |}
    $2 times fact(1)$tikzmark[xshift=0.62em]{1}  hline
    $3 times fact(2)$tikzmark[xshift=0.6em]{2}  hline
    $4 times fact(3)$ hline
 end{tabular}
 &  
 begin{tabular}{ | l |}
    $1 times fact(0)$tikzmark[xshift=0.62em]{3}  hline
    $2 times fact(1)$tikzmark[xshift=0.62em]{4}  hline
    $3 times fact(2)$ hline
    $4 times fact(3)$ hline
 end{tabular}
 & 
 begin{tabular}{ | l |}
    $fact(0) = 1$tikzmark[xshift=0.62em]{5}  hline
    $1 times fact(0)$tikzmark[xshift=0.7em]{6}  hline
    $2 times fact(1)$ hline
    $3 times fact(2)$ hline
    $4 times fact(3)$ hline
 end{tabular} hline
end{tabular}
end{table}
drawcurvedarrow[bend right=50,-stealth]{f}{d}
drawcurvedarrow[bend right=50,-stealth]{2}{1}
drawcurvedarrow[bend right=50,-stealth]{4}{3}
drawcurvedarrow[bend right=50,-stealth]{6}{5}

end{document}

documentclass{article}
usepackage{amsmath}
usepackage{tikz}

newcommand{tikzmark}[2][-3pt]{tikz[remember picture, overlay, baseline=-0.5ex]node[#1](#2){};}

tikzset{brace/.style={decorate, decoration={brace}},
 brace mirrored/.style={decorate, decoration={brace,mirror}},
}

newcounter{brace}
setcounter{brace}{0}
newcommand{drawbrace}[3][brace]{%
 refstepcounter{brace}
 tikz[remember picture, overlay]draw[#1] (#2.center)--(#3.center)node[pos=0.5, name=brace-thebrace]{};
}

newcounter{arrow}
setcounter{arrow}{0}
newcommand{drawcurvedarrow}[3][]{%
 refstepcounter{arrow}
 tikz[remember picture, overlay]draw (#2.center)edge[#1]node[coordinate,pos=0.5, name=arrow-thearrow]{}(#3.center);
}

newcommand{annote}[3][]{%
 tikz[remember picture, overlay]node[#1] at (#2) {#3};
}

begin{document}
begin{table}
centering
begin{tabular}{|c|c|c|c|}
hline
 begin{tabular}[b]{ | l |}
    $3 times text{fact}(2)$tikzmark[xshift=0.62em]{d}  hline
    $4 times text{fact}(3)$tikzmark[xshift=0.6em]{f} hline
 end{tabular}
 &
 begin{tabular}[b]{ | l |}
    $2 times text{fact}(1)$tikzmark[xshift=0.62em]{1}  hline
    $3 times text{fact}(2)$tikzmark[xshift=0.6em]{2}  hline
    $4 times text{fact}(3)$ hline
 end{tabular}
 &
 begin{tabular}[b]{ | l |}
    $1 times text{fact}(0)$tikzmark[xshift=0.62em]{3}  hline
    $2 times text{fact}(1)$tikzmark[xshift=0.62em]{4}  hline
    $3 times text{fact}(2)$ hline
    $4 times text{fact}(3)$ hline
 end{tabular}
 &
 begin{tabular}[b]{ | l |}
    $text{fact}(0) = 1$tikzmark[xshift=0.62em]{5}  hline
    $1 times text{fact}(0)$tikzmark[xshift=0.7em]{6}  hline
    $2 times text{fact}(1)$ hline
    $3 times text{fact}(2)$ hline
    $4 times text{fact}(3)$ hline
 end{tabular} hline
end{tabular}
end{table}
drawcurvedarrow[bend right=50,-stealth]{f}{d}
drawcurvedarrow[bend right=50,-stealth]{2}{1}
drawcurvedarrow[bend right=50,-stealth]{4}{3}
drawcurvedarrow[bend right=50,-stealth]{6}{5}

end{document}