
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 factory, based on past production and sales experience, has obtained the following statistical law regarding production and sales: for every product $x$ (in hundreds of units) produced, the total cost is $G(x)$ (in ten thousand yuan), where the fixed cost is 2.8 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 
$$
R(x)= 
\begin{cases} 
-0.4x^{2}+3.4x+0.8, & (0\leqslant x\leqslant 5) \\
9, & (x > 5)
\end{cases}
$$
Assuming that the products produced are sold (i.e., production equals sales), based on the above statistical law, please solve the following questions:  
(1) Write the expression for the profit function $y=f(x)$ (profit = sales revenue - total cost);  
(2) To make a profit for the factory, find the range of production $x$;  
(3) How many units should the factory produce to maximize profit?

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