Chemotherapy Followed by Peripheral Stem Cell Transplantation Plus Biological Therapy in Treating Women With Stage IV Breast Cancer

DISEASE CHARACTERISTICS: * Women with histologically documented metastatic carcinoma of the breast * Bilateral disease allowed * Concurrent intraductal or lobular carcinoma in situ allowed * Measurable or evaluable recurrent metastatic disease (stage IV) documented by radiograph, CT scan, nuclear medicine scan, or physical exam * Biopsy of recurrent site(s) recommended but not required * Nonmeasurable disease allowed if tumor or metastatic disease has been previously removed or successfully treated * 0 to 3+ HER2 amplification, as determined by FISH * No clinical evidence of active brain metastases * Patients with treated brain metastases (i.e., those who have received definitive radiation, chemotherapy, and/or underwent surgery) and are stable are eligible * Hormone receptor status: * Estrogen or progesterone receptor positive or negative PATIENT CHARACTERISTICS: * Menopausal status not specified * Karnofsky performance status 70-100% OR ECOG performance status 0-2 * Life expectancy at least 3 months * Granulocyte count at least 1,500/mm\^3 * Platelet count at least 50,000/mm\^3 * Hemoglobin greater than 8 g/dL * Bilirubin less than 1.5 times normal * AST, ALT, and alkaline phosphatase \< 5 times upper normal * Creatinine less than 1.8 mg/dL * Creatinine clearance at least 60 mL/min * BUN less than 1.5 times normal * No myocardial infarction (MI) within the past year * No history of MI (\> 1 year ago) with current coronary symptoms requiring medication * No current history of angina/coronary symptoms requiring medication * No clinical evidence of congestive heart failure requiring medical management * No significant congestive heart failure * No other uncontrolled or significant cardiovascular disease * Ejection fraction at least 45% at rest by MUGA * Systolic BP \< 130 mm Hg and diastolic BP \< 80 mm Hg * BP must be controlled to meet the standard by anti-hypertensive medications for at least 7 days prior to the first infusion * PFT-FEV\_1 at least 50% predicted * DLCO2 at least 50% predicted * FVC at least 50% predicted * No other malignancy within the past 3 years * No other serious medical or psychiatric illness that would preclude study participation * HIV negative * Not pregnant or nursing * Negative pregnancy test * Fertile patients must use effective contraception PRIOR CONCURRENT THERAPY: * See Disease Characteristics * Prior chemotherapy regimens allowed, including prior treatment on protocol WSU-2006-130 * Prior vaccine therapy on protocol WSU-2006-130 allowed * More than 4 weeks to leukapheresis since prior hormonal therapy * No radiation to the axial skeleton within 4 weeks of leukapheresis * No concurrent hormonal therapy for breast cancer * Hormones administered for non-disease-related condition (e.g. insulin for diabetes) allowed * Concurrent steroids for adrenal failure, septic shock, or pulmonary toxicity allowed