To solve a problems in probability, a students has $3$ different approaches they can:

$-$ Find the solution directly: this takes $20$ minutes;

$-$ Make a quick mistake: work $10$ minutes and realize it will not work

$-$ the painful mistake, where the student spends one hour $(\backslash (\backslash$mathbf$\{60\}\backslash )$ minutes$)$ computing various integrals before realizing it will not work.

We assume that:

$($a$)$ the student is perseverant: they will not give up until they solve the problem

$($b$)$ they have no memory: they choose one of the three approaches uniformly at random regardless of past failed attempts $($i$.$e$.,$ each approach is used with probability $\backslash (1/3\backslash ),$ and they may try the same erroneous approach even if they just tried and it failed$).$

Compute the expected time for solving the problem.