
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
Given parallelogram \(ABCD\) has an area of \(500,\!000\). The 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 the number of such parallelograms that can be formed given these conditions.

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