
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 subsidiary company of a certain group had a capital of 50 million yuan at the beginning of the year 2023. According to past experience, if all the capital is invested in production, the company's annual capital growth rate is 50%. The group now requires the company to remit 10 million yuan at the end of each year starting from 2023, and invest the remaining capital in production for the next year. Let $a_{n}$ be the remaining capital of the company after remitting the capital at the end of the $n$th year.
  
$(1)$ Find $a_{n}$;  
$(2)$ If the remaining capital of the company at the end of the $m$th year exceeds 30 million yuan, find the minimum value of $m$.

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