
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]


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A company produces ceramics. According to the historical data, the fixed cost of producing ceramics per day is 14,000 yuan, and the cost increases by 210 yuan for each product produced. The daily sales volume $f(x)$ is related to the production quantity $x$ pieces by the following formula: $f\left(x\right)=\begin{cases} \dfrac{1}{625}{x}^{2}, & 0\leqslant x\leqslant 400 \\ 256, & x > 400\end{cases}$. The selling price per product $g(x)$ is related to the production quantity $x$ by the following formula: $g\left(x\right)=\begin{cases}- \dfrac{5}{8}x+750, & 0\leqslant x\leqslant 400 \\ 500, & x > 400\end{cases}$.

    (1) Write down the relationship between the daily sales profit $Q(x)$ and the production quantity $x$ for the ceramic factory;

    (2) To maximize the daily sales profit, how many products should the company produce per day, and what is the maximum profit?

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