'Draw me until you match me' II

Draw me until you match me

This is a question called 'Draw me until you match me', which was posted this week. Well, since I'm not that good at English, I mistakenly misinterpreted the question that the process is repeated until two same numbers were written down 'in a row'. ex: 2,1,3,5,3,1,'2,2' Well, even after I knew I misinterpreted the question and eventually solved it, I kept wondering whether this problem can be solved. (I had a hard time trying to solve this problem..)

Is there a way to solve this question when we changed the question to the one shown above?

Easy Math Editor

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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

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Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$