根据材料完成下列各题。

Beethoven
Born in 1770 in Germany
A  musician (音乐家)
Died in 1827

Florence Nightingale
Born in Italy on May 12th, 1820
An English nurse
Died in 1910

Bill Gates
Born in I955 in the USA
Wrote Business @ the Speed of Thought
Owns Microsoft

Thomas Edison
Born in 1847 in the USA
Made over 1,000 inventions (发明)
Died in 1931

Yao Ming
Born on September 12th, 1980 in Shanghai, China
A basketball player
Played for the Houston Rockets, the USA

已知A（x₁，y₁），B（x₂，y₂）是函数$f\left(x\right)=\left\{\begin{array}{c}\frac{2x}{1-2x},x\ne \frac{1}{2}\\ -1,x=\frac{1}{2}\end{array}\right.$的图象上的任意两点（可以重合），点M在直线x=$\frac{1}{2}$上，且$\overrightarrow{AM}$=$\overrightarrow{MB}$．

（Ⅰ）求x₁+x₂的值及y₁+y₂的值

（Ⅱ）已知S₁=0，当n≥2时，S_n=$f\left(\frac{1}{n}\right)$+$f\left(\frac{2}{n}\right)$+$f\left(\frac{3}{n}\right)$+…+$f\left(\frac{n-1}{n}\right)$，求S_n；

（Ⅲ）在（Ⅱ）的条件下，设a_n=$2^{S_n}$，T_n为数列{a_n}的前n项和，若存在正整数c、m，使得不等式$\frac{T_m-c}{T_{m+1}-c}$成立，求c和m的值．