
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
Let \( T \) denote the value of the sum
\[
\sum_{n=1}^{10000} \frac{1}{\sqrt{n + \sqrt{n^2 + 1}}}
\]
Express \( T \) in the form \( p + q \sqrt{r} \), where \( p, q, \) and \( r \) are positive integers with \( r \) not divisible by the square of any prime. Determine \( p + q + r \).

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