Can I use method of images for a point charge outside a solid dielectric sphere?

My question arised while solving Classical Electrodynamics(J. D. Jackson) Pro 4.9a

Find a potential at all points in space for a point charge $q$ located in a free space a distance $d$ from the center of a solid dielectric sphere of radius $a(a < d)$ and dielectric constant $\frac{\epsilon}{\epsilon_0}$.

My second trial after failure of using image charge method was

1. to divide the space into two regions - inside(1) and outside(2) the sphere

2. In (1) use a solution of Laplace equation($\sum_{l=0}^{\infty} A_l r^l P_l(\cos{\theta})$), in (2) superpose $\frac{q}{4\pi\epsilon_0r}$ and solution of Laplace equation($\sum_{l=0}^{\infty} B_l \frac{1}{r^{l+1}} P_l(\cos\theta)$).

3. Get the expression of $A_l$ and $B_l$ using boundary condition at sphere shell$(r = a)$.

Resulting potential is

At (1) :$\quad \Phi(x) =\frac{q}{4\pi\epsilon_0d} + \sum_{l=1}^{\infty}\frac{q}{4\pi d^{l+1}}\frac{(2l+1)/l}{\epsilon+\epsilon_0\frac{l+1}{l}}r^lP_l(\cos\theta) \quad \cdots(*)$

At (2) :$\quad \Phi(x) = \frac{q}{4\pi\epsilon_0} \frac{1}{|x-d\hat{z}|} + \sum_{l=0}^{\infty}\frac{q}{4\pi d^{l+1}}(1-\frac{\epsilon}{\epsilon_0})\frac{a^{2l+1}}{\epsilon+\epsilon_0\frac{l+1}{l}}\frac{1}{r^{l+1}}P_l(\cos\theta)\quad \cdots(*)$

My first attempt was to locate image charge($q'$) at (2)($x'$) for potential at (1) and use different charge($q''$) at the same location($d\hat z$) for potential at (2), i.e.,

At (1) :$\quad\Phi(x) = \frac{1}{4\pi\epsilon}\frac{q''}{|x-d\hat z|}$

At (2) :$\quad\Phi(x) = \frac{1}{4\pi\epsilon_0} \left(\frac{q}{|x-d\hat z|} + \frac{q'}{|x-x'|}\right)$

and use boundary condition to determine $q',q''$, and $x'$ .

But this turned out to be not working - there was no solution to equations from boundary conditions.

And as a proof, the potential(*) at (2) doesn't seem to be decomposed into two point charges, and potential at (1) doesn't seem to be unified with rescaling of $q$ to $q''$.

This implied either that I used wrong setting for image charge method - number of image charges, location - or that image charge method is not applicable to this problem.

Here is my question.

Is it possible to use method of image in this specific example? I hope the answer to be considering the physical situation of this problem - polarization due to a nearby point charge.

And if some problems don't allow using this approach, which situation allows using this method?

Thank you in advance.