Advanced FILTER Function to Extract Data Subject to Conditions

Hi,

In A1:K16 I have sample data from which I want to extract the results in A22:K27.

A product code will only have a row in the results if the sign (positive or negative) of the value in column F differs from that in column G AND/OR the sign of the value in column J differs from that in column K.

For example:

Product AAA is included in the results because the sign of its value in F2 is positive while its value in G2 is negative
Product EEE is included in the results because the sign of its value in J6 is negative while its value in K6 is positive.
H22:K22 and D23:G23 are blank for AAA and EEE, respectively, because the sign of their values in J2 and K2 and in F6 and G6 are the same.
BBB is not among the results as both of the sign of its values in F3 and G3 and in J3 and K3 are the same.

Can someone please suggest a FILTER function formula to achieve the results in A22:K27 (or some other solution)?

Thanks!