Special 7

A positive integer with seven digits is called special, if it is divisible by 7 and by deleting any one of its digits (exactly one), the number is also divisible by 7.

How many special numbers are there?

\[\begin{align} 7 & \mid \overline{abcdefg} \\ 7 & \mid \overline{abcdef} \\ 7 & \mid \overline{abcdeg} \\ 7 & \mid \overline{abcdfg} \\ 7 & \mid \overline{abcefg} \\ 7 & \mid \overline{abdefg} \\ 7 & \mid \overline{acdefg} \\ 7 & \mid \overline{bcdefg} \end{align}\]

Note: Only the first digit needs to be non-zero. For example 7000000 is a special number, because 7000000, 700000 and 000000 are all divisible by 7.

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