
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 $f(x)=\log_{2}(x+t)$, and $f(0)$, $f(1)$, $f(3)$ form an arithmetic sequence. Let point $P$ be any point on the graph of $y=f(x)$, and let the symmetric point of $P$ with respect to the origin be $Q$, whose trajectory is the graph of the function $y=g(x)$.  
$(1)$ Solve the inequality $2f(x)+g(x)\geqslant 0$ with respect to $x$;  
$(2)$ When $x\in[0,1)$, it always holds that $2f(x)+g(x)\geqslant m$, find the range of $m$.

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