Nested Cuberoots

Just loved this question.. :D $x+1=\sqrt[3]{(x+1)^3}$ $=\sqrt[3]{x^3+3x^2+\color{#D61F06}{3x+1}}$ $=\sqrt[3]{x^3+3x^2+\sqrt[3]{\color{#D61F06}{(3x+1)^3}}}$ $=\sqrt[3]{x^3+3x^2+\sqrt[3]{27x^3+27x^2+\color{#20A900}{9x+1}}}$ $=\sqrt[3]{x^3+3x^2+\sqrt[3]{27x^3+27x^2+\color{#20A900}{(9x+1)^3}}}$ $=\sqrt[3]{x^3+3x^2+\sqrt[3]{27x^3+27x^2+\sqrt[3]{729x^3+243x^2+\color{#3D99F6}{27x+1}}}}$ $=\cdots$ Put x=3 to get the desired result i.e $\Large\color{#624F41}{\boxed{\color{#69047E}{4}}}=\sqrt[3]{54+\sqrt[3]{972+\sqrt[3]{21870+\sqrt[3]{551124+\sqrt[3]{14526054+\cdots}}}}}$