
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
On a $2 \times n$ grid, some cells are painted black. The coloring is called correct if there are no two adjacent painted cells (adjacent cells share a common side). A coloring where no cell is painted at all is also considered correct.

Let $A_n$ be the number of correct colorings with an even number of painted cells, and $B_n$ be the number of correct colorings with an odd number of painted cells. Determine all possible values of $A_n - B_n$.

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