
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
Triangle \(ABD\) has an area of \(500,\!000\). Vertex \(A\) is at \((0,0)\), and all other vertices are in the first quadrant. Vertex \(B\) is a lattice point on the line \(y = 2x\), and vertex \(D\) is on the line \(y = 3x\). Determine how many such triangles exist given that \(k\) can range from 2 to 4.
A) 49
B) 72
C) 200
D) 784
E) 1024

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