They absolutely get to pick!
https://www.uscis.gov/policy-manual/volume-1-part-i-chapter-...
"“Prosecutorial discretion” is the authority of an agency charged with enforcing a law to decide whether to enforce, or not to enforce, the law against someone. USCIS, like other law enforcement agencies, has prosecutorial discretion and exercises it every day. In the immigration context, the term applies not only to the decision to issue a Notice to Appear (NTA), but also to a broad range of other discretionary immigration enforcement decisions."
2) prosecutors can tell police straight up what they will or won’t prosecute, which affects what crimes cops will investigate or make an arrest for
> The pervasive problems with racism in our criminal justice system has been clear. Black Americans are incarcerated in state prisons at nearly five times the rate of white Americans.
> The systemic racism in the system starts before the first contact and continues through charging decisions, plea deals, conviction, sentencing recommendations, incarceration, release and beyond.