
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 with dimensions $EF = 8, FG = 7 = HE,$ and $GH = 6.$ Circles of radius 4 are centered at $E$ and $F$, and circles of radius 3 are centered at $G$ and $H.$ A circle contained within the trapezoid is tangent to all four of these circles. Find the radius of this inner circle, expressed as $\frac{-a+b\sqrt{c}}{d},$ where $a, b, c,$ and $d$ are positive integers, $c$ is not divisible by the square of any prime, and $a$ and $d$ are relatively prime.

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