14. (6th "Hope Cup" Senior High School Competition) The function $y=f(x)$ is an even function and is a periodic function with a period of 2. When $x \in [2,3]$, $f(x)=x-1$. On the graph of $y=f(x)$, there are two points $A$ and $B$, whose y-coordinates are equal ($A$ is to the left of $B$), and their x-coordinates are in the interval $[1,3]$. The fixed point $C$ has coordinates $(0, a)$, where $a>2$. Find the maximum value of the area of $\triangle ABC$ (expressed in terms of $a$).
14. Since $f(x)$ is a periodic function with a period of 2, and $f(x)=x-1, 2 \leqslant x \leqslant 3$,

when $0 \leqslant x \leqslant 1$, $f(x)=f(x+2)=(x+2)-1=x+1$.
Also, since $f(x)$ is an even function.
when $-1 \leqslant x \leqslant 0$, $f(x)=f(-x)=-x+1$.
Therefore, when $1 \leqslant x \leqslant 2$, $f(x)=f(x-2)=-(x-2)+1=-x+3$.
Let the coordinates of $A$ and $B$ be $\left(x_{A}, y_{A}\right),\left(x_{B}, y_{B}\right),\left(1 \leqslant x_{A}<3\right)$, when $a>3$, $\frac{5-a}{2}<1$, then when $x_{1}=1$, $S_{\triangle M}$ has a maximum value, the maximum value $S_{\text {max }}=a-2$.