
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
At the beginning of the year $2023$, a subsidiary company of a group had a capital of $50$ million yuan. Based on past experience, if all the capital is invested in production, the company's annual capital growth rate is $50\%$. Now, the group requires the company to submit $10$ 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 submitting 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 $300$ million yuan, find the minimum value of $m$.

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