
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
Compute the value of 
\[ T = \sqrt{1 + \dfrac1{1^2} + \dfrac1{2^2} + \dfrac1{3^2}} + \sqrt{1 + \dfrac1{2^2} + \dfrac1{3^2} + \dfrac1{4^2}} + \cdots + \sqrt{1 + \dfrac1{100^2} + \dfrac1{101^2} + \dfrac1{102^2}}. \]
Find $\lfloor T^2 \rfloor$.

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