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Start backup: POST http://teamcity:8111/httpAuth/app/rest/server/backup?includeConfigs=true&includeDatabase=true&includeBuildLogs=true&fileName=<fileName>
where <fileName> in is the prefix of the file to save backup to
Get current backup status (idle/running): GET http://teamcity:8111/httpAuth/app/rest/server/backup
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