
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 $f(x)=\log_{a}x$ and $g(x)=\log_{a}(2x+t-2)$, where $a > 0$ and $a\neq 1$, $t\in R$.

(1) If $0 < a < 1$, and $x\in[\frac{1}{4},2]$ such that $2f(x)\geqslant g(x)$ always holds, find the range of values for the real number $t$;

(2) If $t=4$, and $x\in[\frac{1}{4},2]$ such that 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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