
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
Given the functions $f\_1(x)$, $f\_2(x)$, and $h(x)$, if there exist real numbers $a$ and $b$ such that $h(x) = a \cdot f\_1(x) + b \cdot f\_2(x)$, then $h(x)$ is called the harmonious function of $f\_1(x)$ and $f\_2(x)$.
(1) Given the functions $f\_1(x) = x - 1$, $f\_2(x) = 3x + 1$, and $h(x) = 2x + 2$, determine whether $h(x)$ is the harmonious function of $f\_1(x)$ and $f\_2(x)$. Explain your reasoning.
(2) Given that $h(x)$ is the harmonious function of $f\_1(x) = \log\_3 x$ and $f\_2(x) = \log\_{\frac{1}{3}} x$ with $a = 2$ and $b = 1$, find the range of real number values for $t$ such that the equation $h(9x) + t \cdot h(3x) = 0$ has a solution in $x \in [3, 9]$.

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