Not a Coincidence

\[ 0. \, 0041 \, 0043 \, 0047 \, 0053 \, 0061 \, 0071 \, 0083 \, 0097 \, 0113 \, 0131 \, 0151\dots \]

Above are the first few digits decimal expansion of \(\frac{4099200041}{999700029999}\), written in groups of 4.

You may think that all of these numbers are prime. Unfortunately, this pattern of primes does end eventually. After how many prime numbers does this pattern cease?

Inspired by this and this.
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