Example 3 Given $x \in \mathbf{N}_{+}$, if $x^{7} \equiv 1(\bmod 29)$, find the value of $x$ modulo 29.
Solve and verify that 2 is a primitive root modulo 29.
Since $x^{7}=1(\bmod 29)$ has at most 7 solutions, and it is verified that $2^{4} 、 2^{8} 、 2^{12} 、 2^{16} 、 2^{20} 、 2^{24} 、 2^{28}$ are exactly the different solutions of the equation, the equation has exactly these 7 solutions. Therefore, the required are the values of the first 7 numbers modulo 29.