Using a VOD type HLS output is the best way, since the manifest of a VOD HLS rendition contains a list of all segments and the duration of each segment in the EXTINF tag. Adding EXT-X-PROGRAM-DATE-TIME tags to the manifest may also help you to determine the start time of the live event.
Any other option, such as trying to determine the start and end time based on the MediaLive channel channel start/stop, is not as accurate, since this does not take into account the fact that the source could start minutes if not hours after the channel start.
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