Is this calendar information accurate?

If we had 13 months instead of 12, every month would be exactly 28 days. The 1st of each month always be Monday and the 28th would be Sunday.
Every month would have exactly 28 days instead of 28.257, and we would properly align ourselves with the cycles of the moon.

Is this calendar information accurate?
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We are currently using the Gregorian calendar, introduced by Pope Gregory XIII in 1582.
Could it be time for an update?
Is this calendar information accurate?
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