OlympiadBench / 2283 / Chinara starts with the point (3,5), and applies the following three-step…

Chinara starts with the point (3,5), and applies the following three-step process, which we call P :

Step 1: Reflect the point in the x-axis.

Step 2: Translate the resulting point 2 units upwards.

Step 3: Reflect the resulting point in the y-axis.

As she does this, the point (3,5) moves to (3,−5), then to (3,−3), and then to (−3,−3).

Chinara then starts with a different point S₀. She applies the three-step process P to the point S₀ and obtains the point S₁. She then applies P to S₁ to obtain the point S₂. She applies P four more times, each time using the previous output of P to be the new input, and eventually obtains the point S₆(−7,−1). What are the coordinates of the point S₀ ?

Chinara starts with the point $(3,5)$, and applies the following three-step process, which we call $\mathcal{P}$ :

Step 1: Reflect the point in the $x$-axis.

Step 2: Translate the resulting point 2 units upwards.

Step 3: Reflect the resulting point in the $y$-axis.

As she does this, the point $(3,5)$ moves to $(3,-5)$, then to $(3,-3)$, and then to $(-3,-3)$.

Chinara then starts with a different point $S_{0}$. She applies the three-step process $\mathcal{P}$ to the point $S_{0}$ and obtains the point $S_{1}$. She then applies $\mathcal{P}$ to $S_{1}$ to obtain the point $S_{2}$. She applies $\mathcal{P}$ four more times, each time using the previous output of $\mathcal{P}$ to be the new input, and eventually obtains the point $S_{6}(-7,-1)$. What are the coordinates of the point $S_{0}$ ?