Let $$p,q,r,s$$ represent the following propositions.

$$p:\,\,\,x \in \left\{ {8,9,10,11,12} \right\}$$
$$q:\,\,\,x$$ is a composite number
$$r:\,\,\,x$$ is a perfect square
$$s:\,\,\,x$$ is a prime number

The integer $$x \ge 2$$ which satisfies $$\neg \left( {\left( {p \Rightarrow q} \right) \wedge \left( {\neg r \vee \neg s} \right)} \right)$$ is ______________.

Let $${a_n}$$ be the number of $$n$$-bit strings that do NOT contain two consecutive $$1s.$$ Which one of the following is the recurrence relation for $${a_n}$$?

The coefficient of $${x^{12}}$$ in $${\left( {{x^3} + {x^4} + {x^5} + {x^6} + ...} \right)^3}\,\,\,\,\,\,$$ is _____________.

A function $$f:\,\,{N^ + } \to {N^ + },$$ defined on the set of positive integers $${N^ + },$$ satisfies the following properties: $$$\eqalign{ & f\left( n \right) = f\left( {n/2} \right)\,\,\,\,if\,\,\,\,n\,\,\,\,is\,\,\,\,even \cr & f\left( n \right) = f\left( {n + 5} \right)\,\,\,\,if\,\,\,\,n\,\,\,\,is\,\,\,\,odd \cr} $$$

Let $$R = \left\{ i \right.|\exists j:f\left( j \right) = \left. i \right\}$$ be the set of distinct values that $$f$$ takes. The maximum possible size of $$R$$ is _____________________.