
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
Given the real sequences \(\left\{a_{n}\right\} \) and \(\left\{b_{n}\right\}\) that satisfy:
$$(2+\mathrm{i})^{n}=a_{n}+b_{n} \mathrm{i}(n \in \mathbf{N}),$$
calculate the value of \(\sum_{n=0}^{\infty} \frac{a_{n} b_{n}}{7^{n}}\).

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