Call a string of letters "legal" if it can be produced by concatenating $($running together$)$ copies of the following strings:

$\text{'}v,\text{'}ww{w}^{\text{'}},{?}^{\text{'}}{x}^{\text{'}}yy{y}^{\text{'}}y$ and $\text{'}zzz\text{'}.$

For example, the string $\text{'}v{v}^{\text{'}}\text{'}$ is legal because it can be produced by concatenating $\text{'}{?}^{\text{'}},{?}^{\text{'}}v\text{'}$ and $\text{'}v\text{'},$ but the string $\text{'}x{v}^{\text{'}}\text{'}$ is not legal.

For each integer $n1,$ let $t_n$ be the number of legal strings with $n$ letters. For example, $t_1=1(\text{'}v\text{'}$ is the only the legal string$).$

$t_n=a{t}_{n-1}+b{t}_{n-2}+c{t}_{n-3}$ for each integer $n4$

where $a==$

For each integer $n1,$ let $p_n$ be the number of legal strings with $n$ letters that also read the same right to left as they do left to right $($Iike $\text{'}v{x}^{\text{'}},$ for example$).$

Which of the following expressions is equal to $p_{101}?$

$t_{50}+3t_{49}$

$p_{100}+p_{99}$

$t_{50}+t_{49}$

$t_{50}+3t_{48}$

$p_{50}+p_{49}$

$t_{50}+t_{48}$

$t_{100}+t_{99}$

$p_{50}+3p_{49}$