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Obsessively (can be consisdered negative) Dictionary.com:

the domination of one's thoughts or feelings by a persistent idea, image, desire, etc.

Single-mindedly (positive) Dictionary.com

having or showing a single aim or purpose

Faithfully (positive) Dictionary.com

steady in allegiance or affection; loyal; constant

Unswervingly (positive) The Free Dictionary

  1. Not veering or turning aside
  2. Constant; steady