Calculate the primary decomposition

Question

Consider the polynomial ring $R=K[x_1,\ldots, x_8]$ over field $K$. Set $\mathfrak{p}_1=(x_1, x_2, x_5, x_6)$, $\mathfrak{p}_2=(x_3, x_4, x_7, x_8)$ and $I=\mathfrak{p}_1\cap \mathfrak{p}_2$, $f=x_1+x_3$. Is there a way to calculate primary decomposition of $f+I$ manually ? Macaulay2 says

i29 :  primaryDecomposition (f+I)

o29 = {ideal (y , y , x , x , x ), ideal (y , y , x , x , x ), ideal (y , y ,
               2   1   3   2   1           4   3   4   3   1           4   3 
      -------------------------------------------------------------------------
                                2
      y , y , x , x , x  + x , x )}
       2   1   4   2   1    3   3

But I don't know how M2 calculate the primary decomposition. I know if $f$ be a monomial ideal then $f+I=(f+\mathfrak{p}_1)\cap (f+\mathfrak{p}_2)$.

Generally, If $I$ be a monomial ideal of $R$ and $f$ an $R/I$-regular sequence. Is there a relation between $\operatorname{Ass}(I) , \operatorname{Ass}(f+I)$?