
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
On August 9, 2022, President Biden of the United States signed the "2022 Chip and Science Act." For China's semiconductor industry, the short-term impact of the "chip act" may be negative, but it is not decisive because it will stimulate stronger outbreak and lasting power of China's independent innovation. A certain company originally had 400 technical personnel, with an average annual investment of $a$ thousand yuan $(a \gt 0)$. Now, in order to increase investment in research and development work, the company divides the original technical personnel into technical personnel and research and development personnel, with $x$ technical personnel $(x\in N$ and $100\leqslant x\leqslant 275)$. After the adjustment, the annual average investment of research and development personnel increases by $\left(4x\right)\%$, and the annual average investment of technical personnel is adjusted to $a(m-\frac{{2x}}{{25}})$ thousand yuan.<br/>$(1)$ If the total annual investment of the adjusted research and development personnel is not less than the total annual investment of the original 400 technical personnel, what is the minimum number of research and development personnel after the adjustment?<br/>$(2)$ In order to motivate the enthusiasm of research and development personnel and maintain the work enthusiasm of technical personnel, the company decides to simultaneously satisfy the following two conditions in terms of investment: ① The total annual investment of research and development personnel is always not less than the total annual investment of technical personnel; ② The annual average investment of technical personnel is always not reduced. Is there a real number $m$ that satisfies the above two conditions? If so, find the range of $m$; if not, explain the reason.

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