It seems you're dealing with elliptic curve cryptography (ECC) where you have points on a curve and scalar multiplication operations. From your provided information, you have points P1, P2, and P3, as well as scalars A1, A2, A3, and A4.

Your hint suggests that A1, A2, and A3 are related to P1, P2, and P3 respectively, in a way that when you subtract 1 from each scalar and subtract that from its corresponding point, adding the results gives you 2. This indicates that P1, P2, and P3 are likely related to A1-1, A2-1, and A3-1 respectively in some manner.

The relationship between P1, P2, and P3 is also given in terms of scalar operations involving A4. From the relationships provided:

P1 + P2 = A4:

If you add point P1 and point P2 together on the elliptic curve, you get the point A4.
P2 - P1 = P3:

If you subtract point P1 from point P2 on the elliptic curve, you get the point P3.
P3 - A4 = 2P2:

If you subtract point A4 from point P3 on the elliptic curve, you get twice the value of point P2.
P1 - (A4 / 2) = (P3 / 2):

If you subtract half of the value of A4 from point P1 on the elliptic curve, you get half of point P3.
 
Given only A4 and these relationships, it's possible to find the coordinates of at least one of the points P1, P2, or P3 by performing scalar operations with A4.

To find the exact coordinates of the points P1, P2, and P3, you can use scalar multiplication operations on a known base point on the elliptic curve, which is usually provided in ECC. By performing scalar multiplication operations with A4 and using the provided relationships, you can compute the coordinates of at least one of the points. Once you have the coordinates of one point, you can use the relationships between the points to deduce the coordinates of the others.