
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
Twenty buckets, each with a volume of \(6 \mathrm{~L}\), are stacked in a pyramid as shown, with ten buckets in the bottom layer, six in the second layer, three in the third layer, and one in the top layer. Each layer is arranged in an equilateral triangle and the buckets are stacked so that each bucket, other than those in the bottom layer, rests on exactly three buckets in the layer below.

Water is poured into the top bucket one litre at a time. After a litre is poured, the water is allowed to settle before another litre is poured. If water is poured into a bucket that is full, the water spills over the rim of the bucket in such a way that exactly one third of the runoff goes into each of the three buckets on which it rests. For example, once the top bucket is full, additional water spills equally into the three buckets in the second-highest layer.

At the first instant when at least one bucket on the bottom layer is full, what is the total amount of water that has been poured into the top bucket?

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