Find the solutions to the system of nonlinear equations given by:
$$\{\begin{array}{l}y=3x+1\\ y=-x^2+1\end{array}$$
Enter your answer as a list of ordered pair. For example: $(2,-5),(5,-2)$

Step 1 ：The system of equations is nonlinear because one of the equations is quadratic. To find the solutions to the system, we need to find the values of x and y that satisfy both equations simultaneously.

Step 2 ：One way to do this is to set the two equations equal to each other and solve for x. Then, we can substitute the x-values into one of the original equations to find the corresponding y-values.

Step 3 ：Let's start by setting the two equations equal to each other: $3x+1=-x^2+1$

Step 4 ：This simplifies to: $x^2+3x=0$

Step 5 ：We can solve this quadratic equation for x by factoring: $x(x+3)=0$

Step 6 ：Setting each factor equal to zero gives us the solutions x = 0 and x = -3.

Step 7 ：Next, we substitute these x-values into the first equation $y=3x+1$ to find the corresponding y-values.

Step 8 ：For x = 0, y = 1. For x = -3, y = -8.

Step 9 ：Final Answer: The solutions to the system of nonlinear equations are $\boxed{{\displaystyle (0,1),(-3,-8)}}$.