
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  $BCDE$  be a trapezoid with  $BE\parallel CD$ ,  $BE = 20$ ,  $BC = 2\sqrt{34}$ ,  $CD = 8$ ,  $DE = 2\sqrt{10}$ . Draw a line through  $E$  parallel to  $BD$  and a line through  $B$  perpendicular to  $BE$ , and let  $A$  be the intersection of these two lines. Let  $M$  be the intersection of diagonals  $BD$  and  $CE$ , and let  $X$  be the intersection of  $AM$  and  $BE$ . If  $BX$  can be written as  $\frac{a}{b}$ , where  $a, b$  are relatively prime positive integers, find  $a + b$ 

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