
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 produces a certain product, and the defective rate $p$ is related to the daily output $x$ (in ten thousand units) as follows: $P= \begin{cases} \dfrac {1}{6-x}, & 0 < x\leqslant c \\ \dfrac {2}{3}, & x > c\end{cases}$ ($c$ is a constant, and $0 < c < 6$). It is known that for every qualified product produced, the profit is $3$ yuan, and for every defective product, the loss is $1.5$ yuan.  
$(1)$ Express the daily profit $y$ (in ten thousand yuan) as a function of the daily output $x$ (in ten thousand units);  
$(2)$ To maximize the daily profit, what should the daily output be? (Note: defective rate $= \dfrac {\text{number of defective products}}{\text{total number of products}} \times 100\%$)

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