
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 $f(x)={\log_{\frac{1}{3}}}({3^x}+1)+\frac{1}{2}abx$ is an even function, and $g(x)={2^x}+\frac{a+b}{2^x}$ is an odd function, where $a$ and $b$ are complex numbers, then $\left(a+b\right)+(a^{2}+b^{2})+(a^{3}+b^{3})+\ldots +(a^{100}+b^{100})=$____.

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