
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 where
\[\log_{10}(2x+2y) = z \text{ and } \log_{10}(x^{2}+2y^{2}) = z+2.\]
Find constants $c$ and $d$ such that for all $(x,y,z) \in T$, the expression $x^{3} + y^{3}$ equals $c \cdot 10^{3z} + d \cdot 10^{z}.$ What is the value of $c+d$?
A) $\frac{1}{16}$
B) $\frac{3}{16}$
C) $\frac{5}{16}$
D) $\frac{1}{4}$
E) $\frac{1}{2}$

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