
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 $EF = 10$, $FG = GH = 7$, and $HE = 8$. Circles of radius 4 are centered at $E$ and $F$, and circles of radius 3 are centered at $G$ and $H$. Determine the radius of a smaller circle, fully enclosed by the trapezoid, that is tangent to all four of these circles. Express this radius as $\frac{-a + b\sqrt{c}}{d}$, where $a, b, c,$ and $d$ are positive integers, $c$ is not divisible by a square of any prime, and $a, b,$ and $d$ are relatively prime. Find $a + b + c + d$.

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