
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  $ABCD$  has side length  $ 1$ . A circle passes through the vertices of the square. Let  $P, Q, R, S$  be the midpoints of the arcs which are symmetrical to the arcs  $AB$ ,  $BC$ ,  $CD$ ,  $DA$  when reflected on sides  $AB$ ,  $B$ C,  $CD$ ,  $DA$ , respectively. The area of square  $PQRS$  is   $a+b\sqrt2$ , where  $a$  and  $ b$  are integers. Find the value of  $a+b$ .
![Image](https://cdn.artofproblemsolving.com/attachments/4/3/fc9e1bd71b26cfd9ff076db7aa0a396ae64e72.png)

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