
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
Let's find the patterns present in the following table:

$$
\begin{aligned}
1 & =\quad 1 \\
2+3+4 & =1+8 \\
5+6+7+8+9 & =8+27 \\
10+11+12+13+14+15+16 & =27+64
\end{aligned}
$$

Write down the common value of both sides as a product of two factors. Try to continue the table with new rows. What will be the next row? Without writing the intermediate rows, give the first, middle, and last members of the left side of the 20th row, and the two members of the right side. Does the equality also hold there?

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