A 28-year-old man presents to the emergency department with acute onset fever (39.5°C), severe headache, neck stiffness, and petechial rash on the trunk and limbs. CSF analysis shows pleocytosis (WBC 1200/μL, predominantly neutrophils), elevated protein (180 mg/dL), and low glucose (28 mg/dL). Gram stain of CSF shows gram-negative diplococci. What is the most appropriate immediate next step in management?