
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
Four distinct points are arranged on a plane such that the segments connecting them have lengths \(a\), \(a\), \(a\), \(a\), \(2a\), and \(c\). The points form a square and a triangle with one of the diagonals of the square serving as a side of the triangle. The length \(c\) is the hypotenuse of a 45°-45°-90° triangle, where the legs are each \(a\sqrt{2}\). What is the ratio of \(c\) to \(a\)?
A) \(1\)
B) \(\sqrt{3}\)
C) \(2\)
D) \(3\)
E) \(\pi\)

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