Does it make sense to put a back button on first page of a wizard?

I have implemented this four-step wizard.

┌───────────┬───────────┬───────────┬───────────┐  
│         X │         X │         X │         X │  
│           │           │           │           │
│      Next │ Back Next │ Back Next │ Back Save │  
└───────────┴───────────┴───────────┴───────────┘

The Next/Save button is disabled until user takes an action.
X in top right corner closes the modal.

Now a task description has changed and it is now required to implement a Back button on the first page that discards and closes the wizard. I am no UX engineer, but I don't feel it's right.

My thinking behind this is that:

  1. Back button should be bound to the context, that is to navigate between the steps and steps only.
  2. If I am on last page and I want to get quickly to first page, I could mistakenly press the back button four times that would cause me to lose the data entered.
  3. I already have a close button and I should not duplicate functionality.

Could a UX engineer explain what is right and the thinking behind this in correct UX PRO terms?

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