
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 $EFGH$ be an isosceles trapezoid, where $EF = 8$, $FG = 6 = HE$, and $GH = 5$. Circles with radii 4 are centered at $E$ and $F$, and circles with radii 3 are centered at $G$ and $H$. Determine the radius $s$ of a smaller circle entirely contained inside the trapezoid and tangent to all four of these exterior circles. Assume $s$ is of the form $\frac{-a + b\sqrt{c}}{d}$, with $a, b, c, d$ being positive integers, where $c$ has no square prime factor, and $a$ and $d$ are coprime. Find $a + b + c + d$.

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