
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
Given a function $f(x)$ defined on $(0,+\infty)$, the average rate of change of $f(x)$ from $x$ to $x+\Delta x$ is $\frac{{f(x+\Delta x)-f(x)}}{{\Delta x}}=\frac{2}{{\sqrt{x+\Delta x}+\sqrt{x}}}-\frac{1}{{x^2}+x\cdot\Delta x}$. Determine the interval where $f(x)$ is monotonically increasing.

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