
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
During the financial crisis, a steel company was left with a surplus of round steel rods, totaling 2009. These rods are now to be stacked together.
(1) If they are stacked to form a vertical cross-section in the shape of an equilateral triangle (with each layer having one more rod than the layer above), and the aim is to have as few remaining rods as possible, how many rods will be left?
(2) If they are stacked to form a vertical cross-section in the shape of an isosceles trapezoid (with each layer having one more rod than the layer above), and the stack is to be no less than seven layers,
(Ⅰ) How many different arrangements are possible?
(Ⅱ) Considering each rod has a diameter of 10 cm, and to prevent safety hazards, the stack must not be higher than 4 meters, which arrangement should be chosen to save the most space?

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