A counterexample of weak type (1,1) maximal function

Let $ f \in L_{\mathrm{loc}}\left(\mathbb{R}^2\right) $,$$\mathcal{R} = \left\{R \subset \mathbb{R}^2 : R \text{ is a rectangle with sides parallel to the coordinate axes} \right\}.$$Define the strong maximal function of $f$ as$$M_s f(x) = \sup_{\substack{R \in \mathcal{R} \\ x \in R}} \frac{1}{|R|} \int_R |f(y)| \, dy, \quad x \in \mathbb{R}^2.$$Prove that $M_s$ is of type $ (p, p) $, for $ 1 < p < \infty $, and explain why $ M_s $ is not of weak type $ (1,1) $.

For the first part, I use the (p,p) property of Hardy-Littlewood maximal function, $$\|Mf\|_{L^p(\mathbb{R}^2)} \leq C_p \|f\|_{L^p(\mathbb{R}^2)}$$Then $$\|M_sf\|_{L^p(\mathbb{R}^2)}\leq\|Mf\|_{L^p(\mathbb{R}^2)}$$So it's true.

But for the second part, I cannot find a counterexample. I tried something like $\chi_{[0,1]^2}$ but failed.