
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 following number table. The first row contains the numbers from 1 to \( n \). The second row contains the numbers from 1 to \( n \) each multiplied by 2. The rows follow this pattern until the last row, which presents \( n \) times each number from 1 to \( n \).

We will use it to calculate the value of the expression

\[ 1^{3} + 2^{3} + \ldots + 100^{3} \]

In addition to the table, we will use the fact that

\[ 1 + 2 + \ldots + n = \frac{n(n+1)}{2} \]

(a) Determine the sum of all the numbers in row number \( k \). With this, determine an expression for the sum of all the numbers in the table.

(b) Observe sections in the table by separating it into \( L \)-shaped layers. The numbers in a certain layer \( k \) are: \( k, 2k, \ldots, (k-1)k, k^{2}, (k-1)k, \ldots, 2k, k \). Determine the sum of the numbers in this layer as a function of \( k \).

(c) Adding the results of all the layers, we will arrive at the same result as summing all rows. Combining this information, determine the value of the expression:

\[ 1^{3} + 2^{3} + \ldots + 100^{3} \]

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