Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be defined as
$$
f(x)=\left[\begin{array}{ll}
{\left[e^x\right],} & x<0 \\
a e^x+[x-1], & 0 \leq x<1 \\
b+[\sin (\pi x)], & 1 \leq x<2 \\
{\left[e^{-x}\right]-c,} & x \geq 2
\end{array}\right.
$$
where $a, b, c \in \mathbb{R}$ and $[t]$ denotes greatest integer less than or equal to $t$. Then, which of the following statements is true?
