
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
Once, a king called all his pages and lined them up. He gave a certain number of ducats to the first page, two ducats less to the second page, two ducats less again to the third page, and so on. When he reached the last page, he gave them the corresponding number of ducats, turned around, and distributed the ducats in a similar manner back to the beginning of the line (i.e., the second to last page received two ducats less than the last page, etc.). During this round, the first page received two ducats. One of the pages then discovered that they had 32 ducats.

How many pages could the king have had, and how many ducats could he have distributed in total? Determine all possible solutions.

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