Santiago receives $\$ 120$ per year in simple interest from three investments. Part is invested at $2 \%$, part at $3 \%$, and part at $4 \%$. There is $\$ 500$ more invested at $3 \%$ than at $2 \%$. The amount invested at $4 \%$ is two times the amount invested at $3 \%$. Find the amount invested at each rate. The amount invested at $2 \%$ is $\$ \square$ The amount invested at $3 \%$ is $\$$ The amount invested at $4 \%$ is $\$ \square$.

Step 1 ：Let's denote the amount invested at 2% as x, at 3% as y, and at 4% as z.

Step 2 ：From the problem, we know that the total interest earned from all investments is $120. This can be represented as \(0.02x + 0.03y + 0.04z = 120\).

Step 3 ：There is $500 more invested at 3% than at 2%. This can be represented as \(y = x + 500\).

Step 4 ：The amount invested at 4% is two times the amount invested at 3%. This can be represented as \(z = 2y\).

Step 5 ：Solving this system of equations, we find that \(x = 500\), \(y = 1000\), and \(z = 2000\).

Step 6 ：Final Answer: The amount invested at 2% is \(\boxed{500}\), the amount invested at 3% is \(\boxed{1000}\), and the amount invested at 4% is \(\boxed{2000}\).