
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
The length and width of a rectangle are $x$ centimeters and $y$ centimeters, respectively $(x$ and $y$ are positive integers). If the length and width of the rectangle are each increased by $5$ centimeters to form a new rectangle with an area denoted as $S_{1}$, and if the length and width of the original rectangle are each decreased by $2$ centimeters to form a new rectangle with an area denoted as $S_{2}$, find the original rectangle's perimeter if $S_{1}$ is $196$ square centimeters larger than $S_{2}$.

Also, explain why the difference between $S_{1}$ and $S_{2}$ must be a multiple of $7$.

If a rectangle with an area of $S_{1}$ can be combined seamlessly with the original rectangle without any gaps or overlaps to form a new rectangle, write down the relationship between $x$ and $y$ directly.

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