A problem on moderately fast algorithms

For a function $${f(n)}=n\times \lg\left(n\right)$$ and time $$\text{t}=10^{3}$$ $$\mu\text{s}$$, calculate the largest size of the problem $$\text{n} \in \mathbb{N}$$ that can be solved in $$\text{t}$$, assuming that for the algorithm to solve the problem it takes $${f(n)}$$ microseconds.

Details and assumptions:

$$1$$) Do it $$\color{blue}{\text{without WA}}$$

$$2$$) Inspired from $$\color{tan}{\text{CLRS}}$$

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