Problem (Combinatorics):
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 $S_{0}$ ?

Solution:
Suppose that $S_{0}$ has coordinates $(a, b)$.

Step 1 moves $(a, b)$ to $(a,-b)$.

Step 2 moves $(a,-b)$ to $(a,-b+2)$.

Step 3 moves $(a,-b+2)$ to $(-a,-b+2)$.

Thus, $S_{1}$ has coordinates $(-a,-b+2)$.

Step 1 moves $(-a,-b+2)$ to $(-a, b-2)$.

Step 2 moves $(-a, b-2)$ to $(-a, b)$.

Step 3 moves $(-a, b)$ to $(a, b)$.

Thus, $S_{2}$ has coordinates $(a, b)$, which are the same coordinates as $S_{0}$.

Continuing this process, $S_{4}$ will have the same coordinates as $S_{2}$ (and thus as $S_{0}$ ) and $S_{6}$ will have the same coordinates as $S_{4}, S_{2}$ and $S_{0}$.

Since the coordinates of $S_{6}$ are $(-7,-1)$, the coordinates of $S_{0}$ are also $(-7,-1)$.

We work backwards from $S_{6}(-7,-1)$.

To do this, we undo the Steps of the process $\mathcal{P}$ by applying them in reverse order.



Since Step 3 reflects a point in the $y$-axis, its inverse does the same.

Since Step 2 translates a point 2 units upwards, its inverse translates a point 2 units downwards.

Since Step 1 reflects a point in the $x$-axis, its inverse does the same.

Applying these inverse steps to $S_{6}(-7,-1)$, we obtain $(7,-1)$, then $(7,-3)$, then $(7,3)$.

Thus, $S_{5}$ has coordinates $(7,3)$.

Applying the inverse steps to $S_{5}(7,3)$, we obtain $(-7,3)$, then $(-7,1)$, then $(-7,-1)$.

Thus, $S_{4}$ has coordinates $(-7,-1)$, which are the same coordinates as $S_{6}$.

If we apply these steps two more times, we will see that $S_{2}$ is the same point as $S_{4}$.

Two more applications tell us that $S_{0}$ is the same point as $S_{2}$.

Therefore, the coordinates of $S_{0}$ are the same as the coordinates of $S_{6}$, which are $(-7,-1)$.

Final answer: (-7,-1)