(22) For the sequence $\left\{a_{n}\right\}$, we define $\left\{\Delta a_{n}\right\}$ as the first-order difference sequence of $\left\{a_{n}\right\}$, where $\Delta a_{n} = a_{n+1} - a_{n} \left(n \in \mathbf{N}^{*}\right)$. For a positive integer $k$, we define $\left\{\Delta^{k} a_{n}\right\}$ as the $k$-th order difference sequence of $\left\{a_{n}\right\}$, where $\Delta^{k} a_{n} = \Delta^{k-1} a_{n+1} - \Delta^{k-1} a_{n} = \Delta\left(\Delta^{k-1} a_{n}\right)$.
(1) If the sequence $\left\{a_{n}\right\}$ has the first term $a_{1} = 1$, and satisfies
$$
\Delta^{2} a_{n} - \Delta a_{n+1} + a_{n} = -2^{n},
$$

find the general term formula of the sequence $\left\{a_{n}\right\}$;
(2) For the sequence $\left\{a_{n}\right\}$ in (1), does there exist an arithmetic sequence $\left\{b_{n}\right\}$ such that
$$
b_{1} \mathrm{C}_{n}^{1} + b_{2} \mathrm{C}_{n}^{2} + \cdots + b_{n} \mathrm{C}_{n}^{n} = a_{n}
$$

holds for all positive integers $n \in \mathbf{N}^{*}$? If it exists, find the general term formula of the sequence $\left\{b_{n}\right\}$; if not, explain the reason;
(3) Let $c_{n} = (2 n - 1) b_{n}$, and set
$$
T_{n} = \frac{c_{1}}{a_{1}} + \frac{c_{2}}{a_{2}} + \cdots + \frac{c_{n}}{a_{n}},
$$

if $T_{n} < M$ always holds, find the smallest positive integer $M$.