Please help me figure this out. Below is a turing machine that accepts a string with an equal number of a$\text{'}$s b$\text{'}$s and c$\text{'}$s in any order. But, the answer you guys gave below does not accept all string. For instance, the string acbcba is not accepted due to us encountering the empty string at states q$2$ and q$4.$ When I try adding transitions for empty string, it starts accepting strings that are not in the language such as bcca and abcc. Please draw the automaton and update the previous answer I have attached so it works for all strings that are in the language. Procedure:

We have to scan the enput from left to right.

Convert $1^{st}\text{'}a\text{'},1^{st\text{'}\text{'}}b\text{'}$ and $)\text{'}\text{'}$ in the

Scanning to $\text{'}x\text{'},$ then in the $2^{nd}$ turn, Convert

$2^{nd}\text{'}a\text{'}{2}^{nd}\text{'}b\text{'}$ and $2^{nd}\text{'}c\text{'}$ to $x$ and so an$.$

$=>$ we have to repeat this process untill we

Convert all $a^{\text{'}}-s,b^{\text{'}}s$ and $c\text{'}$s to $\text{'}x\text{'}$:

Charecter sconned on between $\text{'}a\text{'},\text{'}b\text{'}$ and $\text{'}c\text{'}$

will not be changed. we can skip those.

Ex:

BBBaabb,$ccccbb$ a $BBB$

$$ Like this way the procedure es going on$.$ If we replece

$x$ for call the letters, that meons the string

as equal noof $a^{\text{'}}-3,$ b$\text{'}$s and c$\text{'}$s$.$

Hence the string is accepted.

$=>$ If the string left with some letters, that means

the string doesn't contain equal noof $a\text{'}$s$,$ b$\text{'}$s

and c$\text{'}$s$.$

Hence the string es not accepted.

Transition Diagram