
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 that the function \( f(x)=\log _{\frac{1}{3}}\left(3^{x}+1\right)+\frac{1}{2} a b x \) is an even function, and \( g(x)=2^{x}+\frac{a+b}{2^{x}} \) is an odd function, where \( a \) and \( b \) are constants, find the value of \( \sum_{k=1}^{2008}\left(a^{k}+b^{k}\right) \).

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