
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]


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Observe the characteristics of the following group of equations and explore the pattern:<br/>①$\sqrt{{1}^{3}}=1=1$;<br/>②$\sqrt{{1}^{3}+{2}^{3}}=1+2=3$;<br/>③$\sqrt{{1}^{3}+{2}^{3}+{3}^{3}}=1+2+3=6$;<br/>④$\sqrt{{1}^{3}+{2}^{3}+{3}^{3}+{4}^{3}}=1+2+3+4=10$.<br/>Based on the pattern of the above equations, answer the following questions:<br/>$(1)1^{3}+2^{3}+3^{3}+4^{3}+5^{3}=( $______)$^{2}=$______;<br/>$(2)\sqrt{{1}^{3}+{2}^{3}+{3}^{3}+...+(n-1)^{3}+{n}^{3}}=\_\_\_\_\_\_;($expressed as an algebraic expression with $n$);<br/>$(3)\sqrt{{1}^{3}+{2}^{3}+{3}^{3}+...+9{9}^{3}+10{0}^{3}}=$______;<br/>$(4)$ Simplified calculation: $11^{3}+12^{3}+13^{3}+\cdots +19^{3}+20^{3}$.

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