= Solution
{c}
Define
$$
L=\{-c-b,\ldots,-c-1\},\quad U=\{c+1,\ldots,c+a\},\quad
C=L\cup I\cup U=\{-c-b,\ldots,c+a\}.
$$
An interior jump cannot go below $-c-b$ or above $c+a$, and every exterior reset lands in $I$. Thus $C$ is closed.
To prove that it is one <communicating class>, start anywhere in $C$. From $L$ or $U$, a reset can reach $I$ with positive <probability>. From any state of $I$, repeated $+a$ steps reach some state of $U$, after staying in $I$ up to the last step. A reset from this state can reach every point of the consecutive block
$$
B=\{c-b+1,\ldots,c\}.
$$
Every <integer> $j\in\{-c-b,\ldots,c\}$ has the form $t-kb$ for some $t\in B$ and an <integer> $k\geq0$. Repeated $-b$ steps from $t$ reach $j$, with all departure states in $I$ until a possible last step into $L$. This gives access to all of $I\cup L$. Finally, each $j\in U$ can be reached from $j-a\in I$. Every path described has strictly positive <probability>. Therefore every state in $C$ reaches every other state in $C$.
Each state outside $C$ can only stay put or reset into $I$, and no transition from $C$ returns to it. Different states outside $C$ cannot reach one another. Hence the complete list of <communicating classes> is
$$
\boxed{C=\{-c-b,\ldots,c+a\}\text{, and }\{i\}\text{ for each }i\notin C.}
$$
The finite class $C$ is the unique <closed communicating class>. The exterior singleton classes are <transient communicating classes>, since their self-transition <probabilities> are strictly less than one. Notice that this reachability proof does not require $a,b$ to be <coprime>.
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