The largest value of non-negative integer $a$ for which $$\lim_{x\to1} \left\{ \frac{-a x+\sin(x-1)+a}{x+\sin(x-1)-1} \right\}^{\frac{1-x}{1-\sqrt{x}}} = \frac{1}{4}$$ is ____.

$\lim_{x\to 1}\left\{\frac{-ax+\sin(x-1)+a}{x+\sin(x-1)-1}\right\}^{\frac{1-x}{1-\sqrt{x}}} = \frac{1}{4}$ $a \in \mathbb{Z}_{\ge 0}$ $\text{Find: the largest non-negative integer } a$ $t = x-1 \Rightarrow x = 1+t, \quad t \to 0$ $\text{Base} = \frac{\sin t - a t}{t + \sin t}$ $\frac{1-x}{1-\sqrt{x}} = \frac{(1-\sqrt{x})(1+\sqrt{x})}{1-\sqrt{x}} = 1+\sqrt{x}$ $\text{so the exponent} \to 2$ $\sin t = t - \frac{t^3}{6} + \cdots$ $\text{Base} = \frac{(1-a)t - \frac{t^3}{6}}{2t - \frac{t^3}{6}} \to \frac{1-a}{2}$ $L = \left(\frac{1-a}{2}\right)^2$ $\left(\frac{1-a}{2}\right)^2 = \frac{1}{4} \Rightarrow (1-a)^2 = 1$ $1-a = \pm 1 \Rightarrow a = 0 \ \text{or} \ a = 2$ $\therefore \text{the largest } a = 2$