<s>[INST] Finding $a + b + c$ given that $\;a + \frac{1}{b+\large\frac 1c} = \frac{37}{16}$ Please help me to find the needed sum:

If $a,b,c$ are positive integers such that $\;a + \dfrac{1}{b+\large \frac 1c} = \dfrac{37}{16},\;$
  find the value of $\;(a+b+c)$.

Thanks!
 [/INST] $$\;\color{blue}{\bf a} + \dfrac{1}{\color{red}{\bf b}+1/\color{green}{\bf c}} = \dfrac{37}{16} =  2 + \dfrac 5{16} = 2 + \dfrac 1{\frac{16}{5}} = \color{blue}{\bf 2} + \dfrac 1{\color{red}{\bf 3} + 1/\color{green}{\bf 5}}$$
Now, match up: $\;\color{blue}{\bf a = \;?}\;\quad \color{red}{\bf b = \;?},\quad \color{green}{\bf c = \;?}$
And then find the sum: $$\quad a + b + c = \;\;?$$
</s>