Step 1 :First, we need to find the common points of the two functions, which means we need to solve the equation $x^{2}-2 x+1=-x^{2}+3 x+4$.
Step 2 :Rearrange the equation, we get $2x^{2}-5x-3=0$.
Step 3 :This is a quadratic equation, and we can solve it by using the quadratic formula $x = \frac{-b \pm \sqrt{b^{2}-4ac}}{2a}$, where $a=2$, $b=-5$, and $c=-3$.
Step 4 :Substitute $a$, $b$, and $c$ into the formula, we get $x = \frac{5 \pm \sqrt{(-5)^{2}-4*2*(-3)}}{2*2}$.
Step 5 :Simplify the equation, we get $x = \frac{5 \pm \sqrt{25+24}}{4}$.
Step 6 :Further simplify the equation, we get $x = \frac{5 \pm \sqrt{49}}{4}$.
Step 7 :So, the solutions are $x = \frac{5+7}{4}$ or $x = \frac{5-7}{4}$.
Step 8 :Simplify the solutions, we get $x = 3$ or $x = -\frac{1}{2}$.
Step 9 :Substitute $x = 3$ and $x = -\frac{1}{2}$ into the first equation $y=x^{2}-2 x+1$, we get $y = 3^{2}-2*3+1$ and $y = (-\frac{1}{2})^{2}-2*(-\frac{1}{2})+1$.
Step 10 :Simplify the equations, we get $y = 4$ and $y = \frac{5}{4}$.
Step 11 :So, the common points of the two functions are $(3,4)$ and $(-\frac{1}{2},\frac{5}{4})$.
Step 12 :Check the solutions by substituting them into the second equation $y=-x^{2}+3 x+4$, we find that they satisfy the equation, so the solutions are correct.
Step 13 :Therefore, the solutions to the system of equations are $\boxed{(3,4)}$ and $\boxed{(-\frac{1}{2},\frac{5}{4})}$.