
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 \(EF = 8\), \(FG = 6 = HE\), and \(GH = 5\). 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 \(r\) of this circle, expressed in the form \(\frac{-a+b\sqrt{c}}{d}\), 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. Find \(a+b+c+d\).

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