
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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In 2020, a year full of challenges, but also full of opportunities and potential. The sudden epidemic has brought huge impact and changes to the world, objectively making people pay more attention to the power and potential of technology. A subsidiary of a certain company is engaged in the production of a high-tech product. Assuming that the company had a capital of 50 million yuan at the beginning of the first year, and invested all of it in production, by the end of that year the capital had increased by 50%. It is expected that the annual growth rate of capital in the future will be the same as the first year. The company requires the subsidiary to pay a certain amount of capital t (t ≤ 2500) million yuan at the end of each year starting from the first year, and invest the remaining capital in production for the next year. Let the remaining capital after paying the capital at the end of the nth year be $a_{n}$ million yuan.  
$(1)$ Determine whether $\{a_{n}-2t\}$ forms a geometric sequence? Explain your reasoning.  
$(2)$ If the subsidiary pays capital t = 1500 million yuan at the end of each year, and the remaining capital of the subsidiary at the end of the mth year exceeds 21000 million yuan, find the minimum value of m. ($\lg 2\approx 0.3010, \lg 3\approx 0.4771$)

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