Let $\mathcal{C}$ be the set of all twice differentiable functions $f:[0,1] \to \mathbb{R}$ with at least two (not necessarily distinct) zeros and $|f''(x)| \le 1,$ for all $x \in [0,1].$ Find the greatest value of the integral $$\int\limits_0^1 |f(x)| \mathrm{d}x$$when $f$ runs through the set $\mathcal{C},$ as well as the functions that achieve this maximum. Note: A differentiable function $f$ has two zeros in the same point $a$ if $f(a)=f'(a)=0.$