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[ASSESS]

[ADVANCE]

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A private enterprise in Guangdong mainly engages in the processing and production of a certain brand of sports shoes in the United States. Following international conventions, the settlement currency is the US dollar. Based on the statistical analysis of past processing production data, if the amount of the processing product order is $x$ thousand US dollars, the approximate processing fee that can be obtained is $\frac{1}{2}\ln \left(2x+1\right)$ thousand US dollars. Influenced by the monetary policy of the Federal Reserve, as the US dollar depreciates, due to the time required for production processing contracts and delivery of finished products, profits will be lost due to the depreciation of the US dollar by $mx$ thousand US dollars, where $m$ is the depreciation index of the US dollar during that period, $m\in \left(0,1\right)$. Therefore, the actual processing fee obtained is $f\left(x\right)=\frac{1}{2}\ln \left(2x+1\right)-mx$ thousand US dollars.

$(1)$ If the depreciation index of the US dollar during a certain period is $m=\frac{1}{200}$, to ensure that the actual processing fee obtained by the enterprise increases with the increase of $x$, what range should the amount $x$ of the enterprise's processing product order be within?

$(2)$ If the amount of the enterprise's processing product order is $x$ thousand US dollars and the total production cost required is $\frac{1}{20}x$ thousand US dollars, and the processing production capacity of the enterprise is $x\in \left[10,20\right]$ (where $x$ is the amount of the product order), in what range should the depreciation index $m$ of the US dollar be for the enterprise's processing production to not incur losses?

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