
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

From the vertices of the square $A_{1} A_{2} A_{3} A_{4}$, we measure a distance of $x$ in both directions along each side. Connecting the endpoints of these measurements in the same direction consecutively, we form the squares $B_{1} B_{2} B_{3} B_{4}$ and $C_{1} C_{2} C_{3} C_{4}$, respectively. The sides of these two squares intersect at the vertices of the octagon $D_{1} D_{2} \ldots D_{8}$.

1. For what value of $x$ will this octagon be regular?

2. The vertices of this regular octagon lying on the diagonals of the original square determine another square. Performing this construction again, another regular octagon is obtained, and so on. What is the sum of the perimeters and the sum of the areas of this infinite sequence of regular octagons?

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