
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 $f(x) = \log_a(x+1)$ ($a>1$), if the trajectory of the point $Q$, which is symmetric to any point $P$ on the graph of the function $y = g(x)$ with respect to the origin, exactly matches the graph of the function $f(x)$.  
(1) Write the analytical expression of the function $g(x)$.  
(2) When $x \in [0,1)$, it always holds that $f(x) + g(x) \geq m$. Find the range of $m$.

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