
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
In timekeeping, the duration of a year is taken as 365 days, 5 hours, 48 minutes, and 46 seconds. Express this in days, convert the measurement to a continued fraction, and then determine the convergents of the continued fraction. Explain the measures taken by Julius Caesar and the Gregorian calendar regarding leap years based on the convergents.

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