
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 components for a type of instrument, and due to limitations in production capacity and technical level, some defective products are generated. According to experience, the defective rate $P$ and the daily output $x$(ten thousand pieces) satisfy the relationship: $P=\begin{cases} \frac{1}{6-x}\;,1\leqslant x\leqslant c \\ \frac{2}{3}\;,x > c \end{cases}\;$(where $c$ is a positive number less than $6$). (Note: defective rate = number of defective products / total production, e.g., $P=0.1$ means that for every $10$ products produced, $1$ is defective, and the rest are qualified products.) It is known that for every ten thousand qualified instruments produced, a profit of $20,000 can be made, but for every ten thousand defective products produced, a loss of $10,000 will be incurred. The factory wants to determine a suitable daily output.

1. Express the daily profit $T$(ten thousand dollars) from producing these instrument components as a function of the daily output $x$(ten thousand pieces);
2. What daily output will yield the maximum profit?

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