
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 $f(x)=\log_a(x+1)$ and $g(x)=2\log_a(2x+t)$, where $t\in\mathbb{R}, a>0$, and $a\neq 1$:  
(Ⅰ) If $3$ is a solution to the equation $f(x)-g(x)=0$, find the value of $t$.  
(Ⅱ) When $0<a<1$ and $t=1$, solve the inequality $f(x) \leqslant g(x)$.  
(Ⅲ) If the function $F(x)=a^{f(x)}+tx^2-2t+1$ has a root in the interval $(-1, 3]$, determine the range of $t$.

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