# Find all integer tuples in a bounded region

For given rational numbers $$p_1/q_1,p_2/q_2,p_3/q_3$$ and a negative integer $$d$$, I'd like to find quadruples of integers $$(w,x,y,z)$$ in a rectangle(of particular sidelengths) satisfying the series of inequalities of the form $$1+w+di+\left[\frac{ip_1+x}{q_1}\right]+\left[\frac{ip_2+y}{q_2}\right]+\left[\frac{ip_3+z}{q_3}\right]\leq 0$$ (for $$1\leq i\leq \mathrm{lcm}(q_1,q_2,q_3)$$). The following is the code that I tried:

Solve[ForAll[i,
i \[Element] Integers &&
1 <= i <= LCM[Denominator[a], Denominator[b], Denominator[c]],
1 + w + i*n + Floor[(i*Numerator[a] + x)/Denominator[a]] +
Floor[(i*Numerator[b] + y)/Denominator[b]] +
Floor[(i*Numerator[c] + z)/Denominator[c]] <= 0] &&
0 <= w <= Floor[2 - n - a - b - c] && 0 <= x < Denominator[a] &&
0 <= y < Denominator[b] && 0 <= z < Denominator[c], {w, x, y,
z}, Integers]


And it shows the message that this system cannot be solved with the methods available to Solve even for a small input like $$(1/2,2/3,4/5,-2)$$. The same happens for other functions with similar roles, like Reduce or FindInstance.

As I'm doing finite search, I suppose there must be a way to make this code work properly in Mathematica. Can someone suggest the correct way to do this?