
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
\[\log_{10}(2x+3y) = z \text{ and } \log_{10}(x^2+2y^2) = z+2.\]
Determine the real numbers $c$ and $d$ such that for all ordered triples $(x,y,z)$ in $T$, it holds that $x^2y + xy^2 = c \cdot 10^{4z} + d \cdot 10^{3z}.$ What is the value of $c+d?$
A) 0
B) $\frac{1}{12}$
C) $\frac{1}{6}$
D) 1

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