
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
It is known that the cost price of type $A$ disinfectant water is $20$ yuan per bottle, and the cost price of type $B$ disinfectant water is $30$ yuan per bottle. A certain distributor purchases type $A$ and type $B$ disinfectant water for $2000$ yuan for resale (both sales volumes are integers). When the selling price of type $A$ disinfectant water is $30$ yuan per bottle, $100$ bottles can be sold. If the price increases by $1$ yuan, the sales volume decreases by $5$ bottles. The selling price of type $B$ disinfectant water is $60$ per bottle, and all purchased type $A$ and type $B$ disinfectant water are sold out.
  
$(1)$ Let the selling price per bottle of type $A$ disinfectant water be $x$ yuan ($x$ is an integer greater than $30$). Fill in the blanks with algebraic expressions containing $x$: ① The sales volume of type $A$ disinfectant water is ______ bottles; ② The total cost price of type $B$ disinfectant water is ______ yuan; ③ The sales volume of type $B$ disinfectant water is ______ bottles.  
$(2)$ Find the maximum value of the total profit from selling type $A$ and type $B$ disinfectant water.  
$(3)$ If the total profit from selling type $A$ and type $B$ disinfectant water is not less than $1945$ yuan, directly write down the possible selling prices per bottle of type $A$ disinfectant water.

Present the answer in LaTex format: \boxed{Your answer}
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