Suppose $\{a_n\}$ is a sequence of real numbers such that $a_1 \neq -3$ and $a_{n+1} = 2a_n + 3$ for each natural number $n \geq 1.$ Which of these statements are true?

A. $\{a_n\}$ is bounded.

B. $\{a_n\}$ is monotone.

C. $\{a_n\}$ converges.

Definitions. A sequence $\{x_n\}$ of real number is:

bounded if there is a real number $M$ such that $-M < x_n < M$ for all $n.$

monotone if $x_n \leq x_{n+1}$ for all $n,$ or $x_n \geq x_{n+1}$ for all $n.$

convergent if there is a number $L$ such that for every $\epsilon > 0,$ there is a natural number $N$ satisfying

$n > N \Rightarrow |x_n - L| < \epsilon.$