
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
Let  $ABCD$  be a unit square. Draw a quadrant of the a circle with  $A$  as centre and  $B,D$  as end points of the arc. Similarly, draw a quadrant of a circle with  $B$  as centre and  $A,C$  as end points of the arc. Inscribe a circle  $\Gamma$  touching the arc  $AC$  externally, the arc  $BD$  externally and also touching the side  $AD$ . Find the radius of  $\Gamma$ .


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