
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
Evaluate the sum $T$ defined by
\[
\sum_{n = 1}^{100} \frac{1}{\sqrt{n + \sqrt{n^2 - 4}}}
\]
and express $T$ if possible in the form $p + q \sqrt{r}$, where $p, q,$ and $r$ are positive integers and $r$ is not divisible by the square of any prime. Find the sum of $p$, $q$, and $r$.

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