
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
In this year's high school entrance examination, a certain school achieved excellent results. In order to share this good news with more people, the school is preparing to print promotional materials. Printing company A proposed: a printing fee of $0.2$ yuan per copy, and an additional plate-making fee of $500; Printing company B proposed: a printing fee of $0.4 yuan per copy, with no plate-making fee.  
$(1)$ Let $x$ represent the number of promotional materials to be printed. Please express with an expression containing $x$: the fee for printing company A is ______ yuan; the fee for printing company B is ______ yuan.  
$(2)$ If the school plans to print 3000 copies of promotional materials, demonstrate through calculation which printing company is more cost-effective.  
$(3)$ Find the value of $x$ for which the costs of both printing companies A and B are the same.

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