
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 sells a children's assembly toy with a cost price of $100$ yuan per unit. During the sales process, it was found that the weekly sales volume $y$ (units) and the selling price per toy $x$ (yuan) satisfy a linear function relationship (where $100\leqslant x\leqslant 160$, and $x$ is an integer). When the selling price per toy is $120$ yuan, the weekly sales volume is 80 units; when the selling price per toy is $140$ yuan, the weekly sales volume is 40 units.  
$(1)$ Find the functional relationship between $y$ and $x$;  
$(2)$ At what price per toy will the e-commerce platform maximize the profit from selling this toy each week? What is the maximum weekly profit in yuan?

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