
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 company has an annual fixed cost of $10$ million yuan for producing a certain brand of clothing. To produce every $1,000$ pieces, an additional cost of $2.7$ million yuan is required. Suppose the company produces $x$ thousand pieces of this brand within a year and sells them all, with revenue of $R(x)$ million yuan per thousand pieces. The revenue function $R(x)$ is defined as:
$$
R(x)= \begin{cases} 
10.8- \dfrac {1}{30}x^{2} & (0 < x\leqslant 10) \\
\dfrac {108}{x}- \dfrac {1000}{3x^{2}} & (x > 10)
\end{cases}
$$
$(1)$ Write the function expression for the annual profit $W(x)$ (in million yuan) as a function of the annual output $x$ (in thousand pieces);
$(2)$ At what annual output (in thousand pieces) does the company maximize its profit from producing this brand of clothing? (Note: Annual profit = Annual revenue - Total annual cost)

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