
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
Let $S$ be the set of ordered triples $(x,y,z)$ of real numbers for which
\[\log_{2}(x+y) = z \text{ and } \log_{2}(x^{2}+y^{2}) = 2z+1.\]
There are real numbers $a$ and $b$ such that for all ordered triples $(x,y,z)$ in $S$ we have $x^{3}+y^{3}=a \cdot 2^{3z} + b \cdot 2^{z}.$ What is the value of $a+b?$
A) 2  
B) 3  
C) \(\frac{5}{2}\)  
D) \(\frac{7}{2}\)

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