
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 certain company's subsidiary is engaged in the production of a kind of high-tech product. At the beginning of the first year, the subsidiary had a capital of 2000 million yuan, which was invested 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 following years will be the same as in the first year. The company requires the subsidiary to pay a certain amount of capital, denoted as \(d\) million yuan, at the end of each year, and to reinvest the remaining capital into production for the next year. Let the remaining capital of the subsidiary at the end of the \(n\)th year, after paying the capital, be \(a_n\) million yuan.

    (1) Express \(a_1\) and \(a_2\) in terms of \(d\), and write the relationship between \(a_{n+1}\) and \(a_n\).

    (2) If the company wishes that after \(m (m \geqslant 3)\) years the subsidiary's remaining capital is 4000 million yuan, determine the value of the capital \(d\) that the subsidiary needs to pay each year (expressed in terms of \(m\)).

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