
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
An e-commerce platform has two live streaming rooms, A and B, with the following promotional activities:<br/>

| Room | Promotion Plan |
|------|-----------------|
| A    | 40% off entire purchase |
| B    | "Get $50 off for every $100 spent" (e.g., buy a $190 item, only pay $140; buy a $200 item, only pay $100) |

Based on the information above, answer the following questions:<br/>
$(1)$ For items priced at $x$ dollars purchased in rooms A and B, the actual payments are $y_A$ and $y_B$ respectively. When $0\leqslant x \lt 200$, find the functional relationship between $y_A$, $y_B$, and $x$;<br/>
$(2)$ For items priced at $x$ dollars in both rooms A and B ($0 \lt x \lt 200$), in what range of values of $x$ is it more cost-effective to buy from room A?

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