
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
Consider the sum of the three tables below. The first represents $n$ rows, with the first row having $n$ numbers all equal to $n$, the second row having $n-1$ numbers all equal to $n-1$, and so on. In the second table, we have a similar distribution of numbers but in columns instead of rows. In the third table, the numbers are in diagonals, with the first diagonal containing one number 1, the second diagonal containing two numbers equal to 2, the third diagonal containing three numbers equal to 3, and so on.

The result of the sum of these three tables will be a table with the same quantity of numbers, with each position being the sum of the corresponding positions in the three tables. For example, in the top left corner, we will have the number $n+n+1=2n+1$.

(a) One way to determine how many numbers there are in each table is to turn one of them upside down and combine it with another to form a rectangle with $n$ rows and double the number of one table. Knowing this, how many numbers are there in one table?

(b) How many times does each number $k$ appear in all three tables?

(c) For each position, row $i$, and column $j$, determine the numbers written in it in the three tables and the resulting table.

(d) Using the information from the previous items, verify that

\[1^{2} + 2^{2} + 3^{2} + \ldots + (n-1)^{2} + n^{2} = \frac{1}{6}n(n+1)(2n+1)\]

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