$```$plaintext

Line $1$:

Counter$1$: Counts boxes $($Trigger on sensor $1)$

Flag$1$: Set when Counter$1=5$

Line $2$:

Counter$2$: Counts boxes $($Trigger on sensor $2)$

Flag$2$: Set when Counter$2=5$

Line $3$:

Counter$3$: Counts boxes $($Trigger on sensor $3)$

Flag$3$: Set when Counter$3=5$

Sequence Control:

Priority Logic to check Flag$1,$ Flag$2,$ Flag$3$

Shaft Control:

If Flag$1$ AND not $($Flag$2$ OR Flag$3)=>$ Open Shaft$1,$ Start Timer$1(6$ sec$)$

If Flag$2$ AND not $($Flag$1$ OR Flag$3)=>$ Open Shaft$2,$ Start Timer$2(6$ sec$)$

If Flag$3$ AND not $($Flag$1$ OR Flag$2)=>$ Open Shaft$3,$ Start Timer$3(6$ sec$)$

Reset:

Timer$1$ expires $=>$ Close Shaft$1,$ Reset Counter$1,$ Reset Flag$1$

Timer$2$ expires $=>$ Close Shaft$2,$ Reset Counter$2,$ Reset Flag$2$

Timer$3$ expires $=>$ Close Shaft$3,$ Reset Counter$3,$ Reset Flag$3$

$```$

Implementation in LOGO! Soft Comfort

$1.$ Create Counters

$-$ Create three counters and link them to the respective input sensors$.$

$2.$ Create Flags

$-$ Use compare blocks to set flags when counters reach $5.$

$3.$ Sequence Logic

$-$ Implement priority logic using AND, OR$,$ and NOT gates to ensure only one shaft operates at a time.

$4.$ Timers

$-$ Use on$-$delay timers to keep the shafts open for $6$ seconds.

$5.$ Output Control

$-$ Link the timer outputs to the respective shaft control relays.

Alternative: Using Other PLCs

If Siemens LOGO! PLC does not meet your requirements due to complexity or speed, consider using more advanced PLCs such as Siemens S$7-1200$ or Allen$-$Bradley MicroLogix which offer more flexibility and faster processing capabilities.

Explanation

$-$ Ensure the sensors and actuators are properly synchronized with the PLC inputs and outputs.

$-$ Fine$-$tune the timers and logic to match the speed of your conveyor system.

$-$ Test the system thoroughly to handle edge cases, such as all lines reaching $5$ boxes simultaneously.

By following this approach, you can ensure smooth and efficient control of your conveyor system using Siemens LOGO! PLC or other suitable PLCs$.$

UPDATE is it possible for you to show me in circuit block diagramm? i need only one picture where its fully done