Kindly answer in full details. Not just some hints.

$($a$)$ In a toxicological study with five treatments T$1,...,$T$5($corresponding to an increasing dose$),$ in pairwise comparisons the following differences are significant: T$1$ vs$.$ T$4,$ T$1$ vs$.$ T$5,$ T$2$ vs$.$ T$4,$ T$2$ vs$.$ T$5,$ T$3$ vs$.$ T$4,$ and T$3$ vs$.$ T$5.$ Perform the insert$-$and$-$absorb algorithm by hand to derive the corresponding letter display.

$($b$)$ Assume that the means of the measured response values are increasing with increasing dose. Sketch a possible profile that fits the situation from $($a$).$ For all doses, draw a boxplot and specify a sample size. Justify your plot.