
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
(1) For  $ a>0,\ b\geq 0$ , Compare

 $ \int_b^{b\plus{}1} \frac{dx}{\sqrt{x\plus{}a}},\ \frac{1}{\sqrt{a\plus{}b}},\ \frac{1}{\sqrt{a\plus{}b\plus{}1}}$ .

(2) Find  $ \lim_{n\to\infty} \sum_{k\equal{}1}^n \frac{1}{\sqrt{n^2\plus{}k}}$ .

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