
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
For any real number $x$, $\lfloor x \rfloor$ represents the largest integer not exceeding $x$. Observe the following equations:

$\lfloor\sqrt{1}\rfloor + \lfloor\sqrt{2}\rfloor + \lfloor\sqrt{3}\rfloor = 3$

$\lfloor\sqrt{4}\rfloor + \lfloor\sqrt{5}\rfloor + \lfloor\sqrt{6}\rfloor + \lfloor\sqrt{7}\rfloor + \lfloor\sqrt{8}\rfloor = 10$

$\lfloor\sqrt{9}\rfloor + \lfloor\sqrt{10}\rfloor + \lfloor\sqrt{11}\rfloor + \lfloor\sqrt{12}\rfloor + \lfloor\sqrt{13}\rfloor + \lfloor\sqrt{14}\rfloor + \lfloor\sqrt{15}\rfloor = 21$

$\dots$

According to this pattern, the $n^{\text{th}}$ equation is $\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_.$

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