
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
A square with side length $a$ is divided by one of its diagonals into two isosceles right triangles. What are the perimeter and area of the quadrilateral whose vertices are: the endpoints of the diagonal and the center of the inscribed circle of one of the isosceles right triangles? (Without using trigonometric tables).

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