Three jobs, A$,$ B$,$ and C$,$ are processed in batches in a manufacturing facility with $3$ machines: M$1,$ M$2,$ and M$3.$ The total number of jobs of type A is $90,$ that of type B is $26$ and that of type C is $24.$ The sequence of operations on the jobs are as follows:

Job A: M$1-$M$2-$M$3$

Job B: M$2-$M$3-$M$1$

Job C: M$1-$M$3-$M$2$

Machine M$3$ has been identified as the bottleneck machine, and the processing schedule on Machine M$3$ reads as follows $($the beginning and ending time of processing of the entire batch $($in minutes$)$ are given$)$:

Job C: $0$ to $240$

Job B: $240$ to $890$

Job A: $890$ to $1790$

If a Shipping buffer of $20$ minutes is introduced, and the transfer batch of job C is $2,$ the processing of job C should start at least by the $\_\_\_\_\_\_\_\_$th minute on M$1$ to ensure that the schedule on the bottleneck machine is followed promptly. Consider the processing time of job C on machine M$1$ as $5$ minutes per job.