
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
The unique ecosystem composed of soil, moisture, and climate in Cangxi has become a paradise for kiwifruit, and is honored by the country as the "number one county of red-heart kiwifruit, the hometown of red-heart kiwifruit." A fruit shop sells red-heart kiwifruit, with an average daily sales of 120 boxes, making a profit of $60$ yuan per box. With the Spring Festival approaching, in order to expand sales, the fruit shop decides to take appropriate price reduction measures. After investigation, it was found that for every $5$ yuan reduction in the price per box of red-heart kiwifruit, the fruit shop can sell an additional 20 boxes on average per day. Let the price reduction per box of red-heart kiwifruit be $x$ yuan.<br/>$(1)$ When $x=10$, find the total profit from selling this red-heart kiwifruit;<br/>$(2)$ Let the total profit from selling this red-heart kiwifruit per day be $w$ yuan.<br/>① Find the analytical expression between $w$ and $x$;<br/>② Determine whether the total profit can reach $8200$ yuan. If it can, find the value of $x$ at this time; if it cannot, find the maximum value of $w$.

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