
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 \( a_{n} = 2^{n}, b^{n} = n (n = 1, 2, 3, \ldots) \), \( A_{n} \) and \( B_{n} \) be the sums of the sequences \( \left\{a_{n}\right\} \) and \( \left\{b_{n}\right\} \), respectively, over the first \( n \) terms. Define \( c_{n} = a_{n} B_{n} + b_{n} A_{n} - a_{n} b_{n} \). Find the sum of the first 10 terms of the sequence \( \left\{c_{n}\right\} \).

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