
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 function $f(x) = \log_3(9^x + 1) - x$.

(1) Determine the even or odd property of the function $f(x)$ and provide a proof;

(2) Let function $g(x) = \log_3(a + 2 - \frac{a+4}{3^x})$. If the inequality $f(x) \geq g(x)$ holds true for $x \in [-1, 1]$, find the range of the real number $a$.

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