Using your Head is Permitted

The license plates in the area where my rush-hour traffic jams were happen to contain no letters. They contain only a string of seven digits. My particular game was to try and represent this string as a concatenation of n<5 odd-length digit palindromes.

A palindrome is a string that can be read backwards to yield the same string as when being read forwards. As an example, "able was I ere I saw elba" is a letter palindrome. An odd-length digit palindrome is a palindromic number which is composed of an odd number of digits in decimal notation.

As you can see, there is no point in creating a 7-digit string as a concatenation of n≥5 odd-length palindromes, as most of the digits end up being part of 1-digit palindromes. 1-digit palindromes are still allowed, but I tried not to overuse them...

In any case, trying out this game, I found out, at first, that only for a very small percentage of the cars was there a winning solution. I therefore added a small "cheat" to the rules. I allowed myself to change one digit to a digit of my choice, before attempting to break down the original number into a concatenation form.

The question this month is this: how many different seven-digit strings exist, which, with the change of at most one digit, can be represented as the concatenation of less than 5, odd-length, palindromic numbers?

Because this question can easily be answered by a computer, there's a twist: answers will be accepted only if they satisfy the following conditions.

1. The result will also be represented as a computation, in addition to its representation as a number.
2. The computation can utilize all of the standard operations, such as the four basic arithmetic operations, powers, root-taking, logarithms and factorials, but shouldn't utilize Σ and Π operations (summation and product-taking of implicit series).
3. The computation's origins (the numbers appearing explicitly in it), must all be in the range of 1-10.
4. The computation must not exceed 25 characters in its length.
5. The answer should be accompanied by a proof, that explains why the computation must yield the correct result. A proof that merely asserts the final numerical outcome, or provides a different computation that reaches it, will not receive credit.

List of solvers: