Marking closed curve using decorations markings without using intersections (TikZ)

I want to mark my closed curves with {Latex[length=4mm, cyan]} like this. Usually, to mark a curve, I use intersections then mark a new custom mark by isosceles triangle and rotate and shift with trial and error. But it’s not efficient. I’d like to use something simple like decorations.marking or else because I just need to adjust the position of arrow and it’s efficient to use. But since my curve is closed, I got a lot of errors when using decorations.markings. Is it possible? Please share your knowledge. Thanks in advance.

Here is my MWE:

documentclass[tikz, border=1mm]{standalone}
usetikzlibrary{arrows.meta, decorations.markings}
begin{document}
    begin{tikzpicture} 
        begin{scope}[>={Stealth[length=4mm]}]
            draw[<->, thick] (-4,0)--(4,0) node[below,shift={(-.2,-.2)}]{Large $x$};
            draw[<->, thick] (0,-4)--(0,4) node[left, shift={(-.1,-.2)}]{Large $y$};
        end{scope}
        begin{scope}[]
            draw[line width=1.5pt] (0,0) circle [radius=3cm];
            draw [red, line width=1.3pt]
            (.1,.4) .. controls (1.2,1.8) and (.4,2.3) .. (-.5,1.9)
            .. controls (-1.2,1.3) and (-1.8,1.5) .. (-2,1.4) 
            to[out=180,in=180] (-2,.2) 
            .. controls (-1.8,.1) and (0,.1) .. (.1,.4) -- cycle;
        end{scope}
    end{tikzpicture}
end{document}

    documentclass[border=1mm]{standalone}
    usepackage[dvipsnames]{xcolor}
    usepackage{tikz}
    usetikzlibrary{arrows.meta, decorations.markings}
    begin{document}
    begin{tikzpicture} 
    begin{scope}[>={Stealth[length=4mm]}]
    draw[<->, thick] (-4,0)--(4,0) node[below,shift={(-.2,-.2)}]{Large $x$};
    draw[<->, thick] (0,-4)--(0,4) node[left, shift={(-.1,-.2)}]{Large $y$};
    end{scope}
    begin{scope}[thick,decoration={
        markings,
        mark=at position 0.35 with {arrow[Cyan]{<}}}]
    draw[line width=1.5pt, postaction={decorate}] (0,0) circle [radius=3cm];
    end{scope}
    begin{scope}[thick,decoration={
        markings,
        mark=at position 0.65 with {arrow[Cyan]{>}}}]
    draw [red, line width=1.3pt, postaction={decorate}]
    (.1,.4) .. controls (1.2,1.8) and (.4,2.3) .. (-.5,1.9)
    .. controls (-1.2,1.3) and (-1.8,1.5) .. (-2,1.4) 
    to[out=180,in=180] (-2,.2) 
    .. controls (-1.8,.1) and (0,.1) .. (.1,.4) -- cycle;
    end{scope}
    end{tikzpicture}
    end{document}