
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 a triangle with  $AB=2$  and  $AC=3$ . Let  $H$  be the orthocenter, and let  $M$  be the midpoint of  $BC$ . Let the line through  $H$  perpendicular to line  $AM$  intersect line  $AB$  at  $X$  and line  $AC$  at  $Y$ . Suppose that lines  $BY$  and  $CX$  are parallel. Then  $[ABC]^2=\frac{a+b\sqrt{c}}{d}$  for positive integers  $a,b,c$  and  $d$ , where  $\gcd(a,b,d)=1$  and  $c$  is not divisible by the square of any prime. Compute  $1000a+100b+10c+d$ .

[i]Proposed by Luke Robitaille

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