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Check whether an alternative firmware already exists (e.g. OpenWrt or DD-WRT).

This is a good article that covers the basics of looking for a UART debug port that could be used as a way to get some low level access to the router and go from there:

https://www.riverloopsecurity.com/blog/2020/01/hw-101-uart/

Try to identify a serial port on the PCB, connect a serial adaptor to it (careful to get the pins and voltage right!) and take it from there

(Edit: or google the PCB board markings, maybe somebody already did this)

Finding the JTAG port is also useful, combined with something like OpenOCD.