
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 $f(x)=\log_{a}x$, $g(x)=\log_{a}(2x+t-2)$, where $a > 0$ and $a \neq 1$, $t \in \mathbb{R}$.  
$(1)$ If $0 < a < 1$, and $x \in \left[ \frac{1}{4}, 2 \right]$ holds, $2f(x) \geqslant g(x)$ always holds, find the range of the real number $t$;  
$(2)$ If $t=4$, and $x \in \left[ \frac{1}{4}, 2 \right]$, the minimum value of $F(x)=2g(x)-f(x)$ is $-2$, find the value of the real number $a$.

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