Write a PDA $($Pushdown Automaton$)$ that accepts the language of strings over $\{$a$,$ b$,$ c$\}$ where the number of instances of ab is equal to the number of instances of bc$.$ This should be done in Mentor Formal Language for PDAs. This is what I currently have:

alphabet: $\{$a$,$ b$,$ c$\}$

start: q$0$

accepting: $\{$q$3\}$

$//$ Initialization

q$0(\_,\_->$ $$,$ q$1)$

$//$ Track $\text{'}$a$\text{'}$ and $\text{'}$b$\text{'}$ for $\text{'}$ab$\text{'}$ sequence

q$1($a$,\_->$ A$,$ q$1)$

q$1($b$,$ A $->$ X$,$ q$1)$

$//$ Track $\text{'}$b$\text{'}$ and $\text{'}$c$\text{'}$ for $\text{'}$bc$\text{'}$ sequence

q$1($b$,\_->$ B$,$ q$1)$

q$1($c$,$ B $->\_,$ q$1)$

$//$ Manage other transitions

q$1($a$,$ X $->$ AX$,$ q$1)$

q$1($b$,$ X $->$ X$,$ q$1)$

$//$ Handle $\text{'}$b$\text{'}$ followed by $\text{'}$c$\text{'}$

q$1($b$,$ X $->$ B$,$ q$2)$

q$2($c$,$ B $->\_,$ q$1)$

$//$ States to handle errors or reject

q$2(\_,\_->\_,$ q$2)$

$//$ Final acceptance condition checking for balanced stack and empty input

q$1(\_,$ $ $->\_,$ q$3)$