
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  $ABC$  be an isosceles triangle with  $AB = AC = 4$  and  $BC = 5$ . Two circles centered at  $B$  and  $C$  each have radius  $2$ , and the line through the midpoint of  $\overline{BC}$  perpendicular to  $\overline{AC}$  intersects the two circles in four different points. If the greatest possible distance between any two of those four points can be expressed as  $\frac{\sqrt{a}+b\sqrt{c}}{d}$  for positive integers  $a$ ,  $b$ ,  $c$ , and  $d$  with gcd $(b, d) = 1$  and  $a$  and  $c$  each not divisible by the square of any prime, find  $a + b + c + d$ .

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