
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 $f(x) = 2\lg(x+1)$ and $g(x) = \lg(2x+t)$ (where $t$ is a constant).  
(1) Find the domain of the function $f(x)$.  
(2) If $x \in [0, 1]$ and $g(x)$ is defined, find the range of the real number $t$.  
(3) If $x \in [0, 1]$ and $f(x) \leq g(x)$ always holds, find the range of the real number $t$.

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