
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 $T$ be the set of ordered triples $(x,y,z)$ of real numbers for which
\[\log_2(x+y) = z \text{ and } \log_2(x^{2}+y^{2}) = z+2.\]
There are real numbers $a$ and $b$ such that for all ordered triples $(x,y,z)$ in $T$ we have $x^{3} + y^{3} = a \cdot 2^{3z} + b \cdot 2^{2z}$. What is the value of $a + b?$
- **A)** $20$
- **B)** $22$
- **C)** $24$
- **D)** $26$
- **E)** $28$

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