
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 had a net profit of 5 million yuan in 2015. Due to reasons such as equipment aging, the company's production capacity is expected to decline year by year. Without technical renovation, it is predicted that the net profit will decrease by 2 million yuan each year starting from 2015. If technical renovation is carried out, a one-time investment of 6 million yuan is required at the beginning of 2016. Excluding the investment for technical renovation, the profit for 2016 is expected to be 7.5 million yuan, and each subsequent year's profit will exceed half of the previous year's profit by 2.5 million yuan.
(1) Let the annual net profit of the company without technical renovation be $a_n$ million yuan for the nth year starting from 2016 (with 2016 as the first year), and the annual profit after technical renovation, excluding the renovation investment, be $b_n$ million yuan. Find $a_n$ and $b_n$.
(2) Let the cumulative net profit without technical renovation be $A_n$ million yuan for the nth year starting from 2016 (with 2016 as the first year), and the cumulative profit after technical renovation be $B_n$ million yuan. Find $A_n$ and $B_n$.
(3) Based on the above prediction, how many years must pass from 2016 for the cumulative profit after technical renovation to exceed the cumulative profit without technical renovation?

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