
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
\[\sum_{n = 1}^{4900} \frac{1}{\sqrt{n + \sqrt{n^2 - 1}}}\]
$T$ can be expressed as $a + b \sqrt{c}$, where $a, b,$ and $c$ are positive integers and $c$ is not divisible by the square of any prime. Determine $a + b + c$.

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