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Are these reduced forms of the same thing?

# Task
You are going to take three strings as input $A$, $B$ and $X$. And your goal is to determine if there exists a third string $S$ such that both $A$ and $B$ can be formed by iteratively removing contiguous substrings of $X$ from $S$. For example if $X = 10101$ then both $10$ and $01$ can be formed from the staring string $S = 1010101$
$$
10\,\,(10101) \rightarrow 10
$$
$$
(10101)\,\,01 \rightarrow 01
$$
Input may be either a list of positive integers or a string of alphanumberic ascii characters. Output should be one of two distinct consistent values for each of the cases. One value should be given if an $S$ exists, and the other if not.
~~This is code-golf the goal is to minimize the size of your source code as measured in bytes.~~
# Test cases
## True
The following cases should give a *true* value. In each I give an example $S$, but this is neither input nor output, it is just provided for demonstration.
```text
A, B, X -> S
10, 0110, 01 -> 0110
~~10 001110, 01 -> 001110~~
10, 01, 10101 -> 1010101
1100, 0101, 10101 -> 11010101101010110101011010101
102, 021, 1021021 -> 1021021021
10, 01, 1010101 -> 101010101
```
## False
```text
A, B, X
100, 01, 10101
10201, 1100, 10
10, 01, 001
10, 01, 1010
```

# Task
You are going to take three strings as input $A$, $B$ and $X$. And your goal is to determine if there exists a third string $S$ such that both $A$ and $B$ can be formed by iteratively removing contiguous substrings of $X$ from $S$. For example if $X = 10101$ then both $10$ and $01$ can be formed from the starting string $S = 1010101$
$$
10\,\,(10101) \rightarrow 10
$$
$$
(10101)\,\,01 \rightarrow 01
$$
Input may be either a list of positive integers or a string of alphanumberic ascii characters. Output should be one of two distinct consistent values for each of the cases. One value should be given if an $S$ exists, and the other if not.
This is code-golf. The goal is to minimize the size of your source code as measured in bytes.
# Test cases
## True
The following cases should give a *true* value. In each I give an example $S$, but this is neither input nor output, it is just provided for demonstration.
```text
A, B, X -> S
10, 0110, 01 -> 0110
10, 001110, 01 -> 001110
10, 01, 10101 -> 1010101
1100, 0101, 10101 -> 11010101101010110101011010101
102, 021, 1021021 -> 1021021021
10, 01, 1010101 -> 101010101
```
## False
```text
A, B, X
100, 01, 10101
10201, 1100, 10
10, 01, 001
10, 01, 1010
```

code-golf string abstract-algebra