
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 $\log_{10}(x+y) = z$ and $\log_{10}(x^{2}+y^{2}) = z+1.$ Express $x^{3}+y^{3}$ in terms of $x+y$ and $x^{2}+y^{2}$, and determine the value of $a$ and $b$ such that $x^{3}+y^{3}= a \cdot 10^{3z} + b \cdot 10^{2z}$ for all ordered triples $(x,y,z)$ in $S.$ Find the value of $a+b.$

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