
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
Given an isosceles right triangle \(ABC\) where \(AB\) is the hypotenuse, we then draw another isosceles right triangle \(BCC_{1}\) on hypotenuse \(BC\). Following this pattern, we draw subsequent isosceles right triangles \(BC_{1}C_{2}, BC_{2}C_{3}\), and so on.

Calculate the length of the broken line \(ACC_{1}C_{2}C_{3}\ldots C_{n}\), as well as the sum of the areas of these triangles, if \(AB = a\). What are the results if \(n\) becomes infinite?

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