$($Project scheduling$)$ You are trying to plan a schedule for completing your course projects next

semester. You will be given $k$ course projects. For each project $i,$ you know the the release time

$r_i,$ the length of time $l_i$ you have to spend on it$,$ and the due date $d_i.$ You would like to determine

whether you can find an uninterrupted schedule where you are working on exactly one project at a

time from the beginning to the end and satisfying all the deadlines. For simplicity, you can think

of all numbers involved as positive integers. For example, if you have Project $1$ released at time $1,$

due at time $11,$ taking $2$ time units, Project $2$ released at time $3,$ due at time $9,$ taking $4$ time units,

and Project $3$ released at time $2,$ due at time $7,$ taking $3$ time units, you can have an uninterrupted

schedule as follows: you do Project $3$ from $2$ to $5,$ Project $2$ from $5$ to $9,$ and Project $1$ from $9$ to

We formalize this problem as a language as follows: $PS=\{\begin{array}{ccc}(\text{:}k,& (r_1,l_1,d_1),\text{d}\text{o}\text{t}\text{s}\text{,}(r_k,l_k,& d_k)\text{:})|\end{array}$ all

numbers in the input are positive integers and there exists an uninterrupted schedule