Prioritize Columns to add non-blank cells per row

I figure there's a pretty simple formula to resolve this.

I would like Excel to Sum only 1 non-blank cell per row, based on a column priority.

Column Priority in the sample below is E/C/D/A/B

So, the process would be:

Excel checks E2 (blank), then moves to C2 (blank), then moves to D2 (3) and stops.
Then in the next row, it checks E3 (blank), then moves to C3 (13) and stops.
In the 3rd row, it checks E4 (8) and stops.
In row 4, it checks E5 (1) and stops.
In row 5, it checks E6 (blank), then moves to C6 (11) and stops.
In row 6, it checks E7 (blank), then C7 (blank), then to D7 (blank), then to A7 (blank), then to B7 (7) and stops.
Excel adds 3+13+8+1+11+7 = 43

a	b	c	d	e
12	10		3
11	6	13
		6	8	8
9	25			1
5	17	11	8
	7

Your help with this is much appreciated!

Why does row 6 not use 23 in column E, which is meant to be top priority?

Can we use helper columns?

Could you answer my other question?

One way with a helper column (could be hidden):

Excel 2016 (Windows) 32 bit

	A	B	C	D	E	F	G	H	I
1	12	10		3		3
2	11	6	13			13		TOTAL:	43
3			6	8	8	8
4	9	25			1	1
5	5	17	11	8		11
6		7				7

Sheet: Sheet2

Excel 2016 (Windows) 32 bit

	F
1	=IF(E1<>"",E1,IF(C1<>"",C1,IF(D1<>"",D1,IF(A1<>"",A1,IF(B1<>"",B1,0)))))

Sheet: Sheet2

Excel 2016 (Windows) 32 bit

	I
2	=SUM(F1:F6)

Sheet: Sheet2

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