The router will have a power input, a line connection and several Ethernet ports.
When it is running and you want to change it out, the best way to do that is to either turn off the power supply at the mains or disconnect the power input. On loss of power, the modem sends a "dying gasp" message to the DSLAM indicating loss of power rather than a line disconnect. Wait around 30 second to 1 minute. and then disconnect the line. As you are changing it out also remove the Ethernet connections.
My preferred way of restarting/reconnecting is to power up the new router and give it two or three minutes to fully boot. When stable, then connect the line wait 30 seconds or so before connecting the Ethernet side.
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