
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
Let $M[A]$ denote the number of elements in the non-empty set $A$. Define $|A - B|$ as follows: $|A - B| = \begin{cases} M[A] - M[B], & M[A] \geq M[B] \\ M[B] - M[A], & M[A] < M[B] \end{cases}$. Given $A = \{1, 2, 3\}$ and $B = \{x \mid |x^2 - 2x - 3| = a\}$, find the range of real values for $a$ such that $|A - B| = 1$.

Present the answer in LaTex format: \boxed{Your answer}
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