
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  $a$  and  $b$  be two positive real numbers, and  $AB$  a segment of length  $a$  on a plane. Let  $C,D$  be two variable points on the plane such that  $ABCD$  is a non-degenerate convex quadrilateral with  $BC=CD=b$  and  $DA=a$ . It is easy to see that there is a circle tangent to all four sides of the quadrilateral  $ABCD$ .
Find the precise locus of the point  $I$ .

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