
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
Alvita is planning a garden patio to be made from identical square paving stones laid out in a rectangle measuring $x$ stones by $y$ stones. She finds that when she adds a border of width one stone around the patio, the area of the border is equal to the original area of the patio. How many possible values for $x$ are there?

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