
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
The circles with centers \( E \) and \( F \) are tangent to segment \( BD \) and to the semicircles with diameters \( AB \), \( BC \), and \( AC \). Let \( r_1 \), \( r_2 \), and \( r \) be the radii of the semicircles with diameters \( AB \), \( BC \), and \( AC \), respectively. The radii of the circles with centers \( E \) and \( F \) are \( l_1 \) and \( l_2 \), respectively.

a) Verify that the distance from \( E \) to segment \( AC \) is \( \sqrt{(r_1 + l_1)^2 - (r_1 - l_1)^2} \).

b) Verify that \( l_1 = \frac{r_1 r_2}{r_1 + r_2} \).

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