
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 certain township has fully implemented the rural revitalization strategy and vigorously promoted the "knitted toy" processing industry. A production cooperative has established a branch for processing knitted toys, which requires a fixed cost investment of $100,000$ yuan per year. For processing $x$ thousand toys, the variable cost is $C\left(x\right)$ thousand yuan. When the processing quantity is less than $15$ thousand toys in a year, $C\left(x\right)=12x-12\ln \left(x+1\right)$; when the processing quantity is not less than $15$ thousand toys in a year, $C\left(x\right)=21x+\frac{256}{x-2}-200$. Through market analysis, the processed toys are purchased by the main factory at a price of $20$ yuan per toy.

$(1)$ Find the analytical expression of the annual profit $f\left(x\right)$ in terms of the annual processing quantity $x$. (Annual profit $=$ annual sales revenue $-$ variable cost $-$ fixed cost)

$(2)$ For how many thousand toys processed in a year will the annual profit of the cooperative be maximized? What is the maximum annual profit? (Reference data: $\ln 2\approx 0.69$)

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