The Largest Shifted Interval

A quadratic function $f(x)=ax^2+bx+c$ ($a,b,c\in\mathbb R$, $a\neq 0$) satisfies the following conditions:

1. $f(x-4)=f(2-x)$ for all $x\in\mathbb R$, and $f(x)\geqslant x$ for all $x\in\mathbb R$; 2. $f(x)\leqslant\left(\dfrac{x+1}2\right)^2$ for all $x\in(0,2)$; 3. the minimum value of $f(x)$ on $\mathbb R$ is $0$.

Find the largest $m$ ($m>1$) such that there exists $t\in\mathbb R$ with