Functions

If a function \(f\) satisfies \(f(\mathrm{~m}+\mathrm{n})=f(\mathrm{~m})+f(\mathrm{n})\) for all \(\mathrm{m}, \mathrm{n} \in \mathbf{N}\) and \(f(1)=1\), then the largest natural number \(\lambda\) such that \(\sum_\limits{\mathrm{k}=1}^{2022} f(\lambda+\mathrm{k}) \leq(2022)^2\) is equal to _________.