
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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In the 7th grade, the members of the Zhiyuan team conducted independent research on mathematical micro-projects, combining the recent content they have learned to study the properties of cubic numbers. Based on the background material, explore and solve the following problem:

Explore the properties of cubic numbers

| Material | Ancient Greek mathematicians discovered: a positive integer $k$ can always be expressed as the sum of $k$ consecutive odd numbers. |
| --- | --- |
| Example | $1^{3}=1$ <br/> $2^{3}=3+5$ <br/> $3^{3}=7+9+11$ <br/> $(1)$ According to the pattern, write down: $4^{3}=\_\_\_\_\_\_.$ |

| Inductive Mathematical Rules | $(2)$ If $k^{3}$ is expressed as the sum of $k$ consecutive odd numbers, and one of the odd numbers is $35$, then $k=$______. <br/> $(3)$ When $k=10$, the middle two numbers on the right side of the equation (i.e., the 5th number and the 6th number) are______. |

| Application of Mathematical Rules | $(4)$ Using this conclusion, calculate: $1^{3}+2^{3}+3^{3}+\ldots +10^{3}+11^{3}$ |

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