Consider a string of \(n\) \(7\)'s, \(7777\cdots77,\) into which \(+\) signs are inserted to produce an arithmetic expression. For example, \(7+77+777+7+7=875\) could be obtained from eight \(7\)'s in this way. For how many values of \(n\) is it possible to insert \(+\) signs so that the resulting expression has value \(7000\)?

(第二十二届AIME2 2004 第14题)