
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
Given a table $A$ consisting of $n \times n$ rational numbers arranged in $n$ rows and $n$ columns, where $a_{ij}(i,j=1,2,3,\ldots ,n)$ represents the number located in the $i$-th row and $j$-th column, and $a_{ij}$ takes the value of $1$ or $-1$. The table $A$ is defined as follows: Let $x_{i}$ be the product of the numbers in the $i$-th row of table $A$, and $y_{j}$ be the product of the numbers in the $j$-th column of table $A$. Define $S=(x_{1}+x_{2}+\ldots +x_{n})+(y_{1}+y_{2}+\ldots +y_{n})$, where $S$ is called the "sum of products" of table $A$.

$(1)$ When $n=4$, for the table $A$ shown below, find the value of the "sum of products" $S$ of this table:
$$
\begin{array}{cccc}
1 & 1 & -1 & -1 \\
1 & -1 & 1 & 1 \\
1 & -1 & -1 & 1 \\
-1 & -1 & 1 & 1 \\
\end{array}
$$

$(2)$ Does there exist a $3 \times 3$ table $A$ such that the "sum of products" $S=0$? Explain your reasoning.

$(3)$ When $n=10$, list all possible values of the "sum of products" $S$ for the table $A$.

Present the answer in LaTex format: \boxed{Your answer}
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