
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 $30^\circ$-$60^\circ$-$90^\circ$ triangle $PQR$ has the hypotenuse $PQ=2$. A circle with center $S$ is tangent to the hypotenuse $PQ$ of triangle $PQR$ and to the positive x-axis and y-axis in the coordinate plane. The center $S$ of this circle lies in the first quadrant. Further, there is a second circle, larger, tangent to the y-axis and sharing the tangent point to the hypotenuse $PQ$ with the first circle. Find the radius of the smaller circle.

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