
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 $f(x)=\log_a{x}$ and $g(x)=2\log_a(2x+t-2)$ where $a>0$, $a \neq 1$, and $t \in \mathbb{R}$.
1. When $t=4$ and $x \in [1,2]$, and $F(x)=g(x)-f(x)$ has a minimum value of 2, find the value of $a$.
2. When $0<a<1$ and $x \in [1,2]$, if $f(x) \geq g(x)$ always holds, find the range of the real number $t$.

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