
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 manufacturer, based on past sales experience, has obtained the following statistical law regarding production and sales: for every $x$ (in hundreds of units) produced, where the fixed cost is $2$ ten-thousand yuan, and the production cost per hundred units is $1$ ten-thousand yuan (total cost = fixed cost + production cost), the sales revenue $R(x)$ (in ten-thousand yuan) satisfies $R(x)=\begin{cases} & -0.4x^{2}+4.2x-0.8 & (0\leqslant x\leqslant 5) \\ & 10.2 & (x > 5) \end{cases}$. Assuming the product reaches the break-even point, based on the above information:

(Ⅰ) Find the analytical expression for the profit function $y=f(x)$ (profit = sales revenue - total cost);

(Ⅱ) How many units should the factory produce to maximize profit? What is the maximum profit?

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