Transform LMI problem into different SDP form

Question

I am trying to transform a LMI problem of the following form:

$ min_x \quad c^Tx \\ s.t. \quad x_1A_1+\dots+x_nA_n \preceq R$

into another SDP formulation:

$ min_x \quad c^Tx \\ s.t. \quad Ax = b, \quad X \preceq 0$

where x=vec(X) obtained by stacking the columns of X. Thank you!

Initial Idea:

1. Introduce slack variable S.
2. Replace LMI constraint by:

$ x_1A_1+\dots+x_nA_n - R = S, \quad S \preceq 0$

3. Augment the decision variable x --> $\tilde{x}=[x,s]^T$.
4. Replace c by $\tilde{c}=[c,0]^T$.

But then I dont know how to continue.