USSR Problem Book, #xx hint

The USSR Olympiad Problem Book

#95.* Let \(\{u_n\}_{n=0}^{\infty} = 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, \cdots\) be the Fibonacci numbers (indexing starts at 0, so \(u_0 = 0, \; u_1 = 1, \; u_2 = 1, \cdots\)). Is there a number terminating with four zeroes among the first 100,000,001 Fibonacci numbers?

Hint. Try the problem with one zero rather than four. Then two, three, and four zeroes. Write a computer program to detect the likely patterns. Prove and use this Fibonacci identity: \[u_{k+j} = u_j \cdot u_{k+1} + u_{j-1} \cdot u_k, \quad k \geq 0, \; j \geq 1.\]