Continuing With Continued Fractions

Calculus Level 2

\[ \begin{array} {l l} 1 & = 3 - \frac{2}{1} \\ & = 3 - \frac{2}{ 3 - \frac{2}{1} } \\
& = 3 - \frac{2}{ 3 - \frac{2} { 3 - \frac{2}{1} } } \\
& = \ldots \\ \end{array} \]

Observe the construction of the continued fraction above.

What is the value of

\[ 3 - \dfrac{2}{ 3 - \dfrac{2} { 3 - \dfrac{2} { 3 - \ddots } } }= \ ? \]


Inspiration, see solution comments

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