如图所示,在平面直角坐标系xoy中,正方形OABC的边长为2cm,点A、C分别在y轴的负半轴和x轴的正半轴上,抛物线经过点A、B和D（4,$-\frac{2}{3}$）．

（1）求抛物线的表达式．
（2）如果点P由点A出发沿AB边以2cm/s的速度向点B运动,同时点Q由点B出发,沿BC边以1cm/s的速度向点C运动,当其中一点到达终点时,另一点也随之停止运动．设S=PQ²（cm²）．
①试求出S与运动时间t之间的函数关系式,并写出t的取值范围;
②当S取$\frac{5}{4}$时,在抛物线上是否存在点R,使得以点P、B、Q、R为顶点的四边形是平行四边形?如果存在,求出R点的坐标;如果不存在,请说明理由．
（3）在抛物线的对称轴上求点M,使得M到D、A的距离之差最大,求出点M的坐标．

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    During the Renaissance, new ideas and values gradually ____ those held in the Middle Ages. People began to ____ less on religious themes and adopt a more humanistic attitude to life. … Rich people wanted to ____ their own paintings, so they could ____ their superb palaces and great houses.