HIV Test at 3 weeks accurate?

so today i got tested for hiv at a planned parenthood facility and it was the rapid test something like that where they prick your finger tip and take a small sample of your blood. its been almost 3 weeks since my last exposure and i havent had any symptom other then paranoia of me getting hiv. the test came back negative and i was somewhat relieved. i was told to go back in about 3-6 months for another test. off topic i was kinda sacred that the nurse used a needle that was previously used by another patient who had hiv. i think its just the paranoia kicking in anyways how accurate is the test at 3 weeks? and is it possible i could of contracted hiv from that needle the nurse used on me to draw blood??