已知向量$\stackrel{\rightharpoonup }{a}=\left(m,\cos 2x\right)$，$\stackrel{\rightharpoonup }{b}=\left(\sin 2x,n\right)$，设函数$f\left(x\right)=\stackrel{\rightharpoonup }{a}·\stackrel{\rightharpoonup }{b}$，且$y=f\left(x\right)$的图象过点$\left(\frac{\mathrm{\pi }}{12},\sqrt{3}\right)$和点$\left(\frac{2\mathrm{\pi }}{3},-2\right)$.
（Ⅰ）求$m,n$的值；
（Ⅱ）将$y=f\left(x\right)$的图象向左平移$\phi$$\left(0<\phi <\mathrm{\pi }\right)$个单位后得到函数$y=g\left(x\right)$的图象.若$y=g\left(x\right)$的图象上各最高点到点$\left(0,3\right)$的距离的最小值为$1$，求$y=g\left(x\right)$的单调增区间.