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The K701 are inefficient low impeadance headphones. So you need portable amps that can provide sufficient current to drive them properly.
At the $300 budget I can think of only 2 amps that can possibly do that-
Ibasso P4+ (Can deliver 1000 mW + 1000 mW @ 32 ohms with the right OPAMP combo) - $245 + shipping
Ibasso PB1 (Can deliver approximately 800 mW + 800 mW @ 32 ohms) - $229 + shipping
The P4+ is probably the better amp since its a pure single ended amp while the PB1 is a balanced amp and you will be running it in single ended mode.
Also the P4 comes with a variety of OPAMPs to change the flavour of the sound. So its very useful to get the sound that you want out of the K701.
Good call if you need portable.