3SAT to 3SAT-WITH-MAJORITY reduction I've been struggling with reduction while solving this exercise and I would really appreciate some help if not the solution.
Problem: [3-SAT-WITH-MAJORITY]
Input: A set of clauses $C = {\{c_1, \dots, c_k}\}$ over $n$ boolean variables $x_1, \dots, x_n$ where every clause contains exactly three literals.
Output: YES if there is a truth assignment to the variable such that both

every clause is satisfied under the assignment
there are at least k/2 clauses in which all 3 literals are set to true under the assignment.

NO otherwise.
Show that this problem is NP-complete.
Hint: Take a SAT instance $C$ with $k$ clauses and introduce $3k$ new variables $\top^i_1, \top^i_2, \top^i_3$ for $i \leq k$ and construct $k$ new clauses $c^\top_i = \{ \top^i_1,\top^i_1,\top^i_3 \}$.  Let $C' = C \cup \bigcup_{i=1}^k c^\top_i$ be the input to 3SAT-WITH-MAJORITY.