There isn't one single cop anywhere in America who can go five seconds without saying something false. That's why I don't understand the marginal utility of more cops in the room. The person you want in the room is your attorney.
Ding ding ding ding. Not just that, but you have to affirmatively and unambiguously invoke your 5th Amendment rights. As in, "I wish to invoke my rights under the 5th Amendment of the United States Constitution to remain silent while my lawyer is not present, and I wish to explicitly invoke my 6th Amendment right to counsel." No weasel words, no maybes or I dunnos. Invoke your rights, and if they persist, invoke your rights again, and again, and again. It's the only way to save your ass before the lawyer arrives because the police are simply seeking to catch somebody, not the right person necessarily, and they will literally outsource the lying to the jailhouse snitch to make it happen, amongst other tricks. You can't explain your way out of a situation where the police feels that you are their best shot at getting a perp walk arrange ASAP. Your lawyer can though.
And goddamn the cops will lie to you about how much worse your life is going to get for invoking the 5th. Most people do not have the experience or fortitude to stand up to an authority figure like this, they've been taught just the opposite, and cops know how to abuse the hell out of it.