
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 its past production and sales experience, has obtained the following statistical laws regarding production and sales: The total cost of producing $x$ (in hundreds of units) is $G(x)$ (in ten thousand dollars), where the fixed cost is $15$ ten thousand dollars, and the production cost for each hundred units is $5$ ten thousand dollars ($(\text{Total Cost} = \text{Fixed Cost} + \text{Production Cost})$). The sales revenue $R(x)$ is given by $R(x) = \begin{cases} -2x^{2}+21x+1 &(0 \leqslant x \leqslant 5) \\ 56 &(x > 5) \end{cases}$. Assuming that the product's production and sales are balanced (i.e., all produced products can be sold), find the following based on the given statistical laws:
(1) Write the profit function $y=f(x)$ analytically $(\text{Profit} = \text{Sales Revenue} - \text{Total Cost})$
(2) Find the maximum profit that Factory A can achieve.

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