
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
A furniture factory produces desks priced at $200$ yuan each and chairs priced at $80 yuan each. During a promotional period, the factory offers two discount options to customers:<br/>Option 1: Buy one desk and get one chair for free;<br/>Option 2: Pay $80\%$ of the total price for both the desk and chair.<br/>A school plans to purchase $100$ desks and $x$ chairs.<br/>$(1)$ If $x=100$, calculate which option is more cost-effective;<br/>$(2)$ If $x \gt 100$, express the costs of both options using an algebraic expression involving $x$;<br/>$(3)$ If $x=300$, and both options can be used simultaneously, help the school design the most cost-effective plan.

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