
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
Let the function $f(x)=\log_{a}(x-3a)$ where $a>0$ and $a\neq 1$. When point $P(x,y)$ is on the graph of $y=f(x)$, point $Q(x-2a,-y)$ is on the graph of $y=g(x)$.  
(1) Write the expression for the function $y=g(x)$.  
(2) If for $x\in[a+2,a+3]$, it always holds that $|f(x)-g(x)|\leq 1$, determine the range of values for $a$.  
(3) By translating the graph of $y=g(x)$ to the left by $a$ units, we obtain the graph of $y=h(x)$. The function $F(x)=2a^{1-h(x)}-a^{2-2h(x)}+a^{-h(x)}$ (where $a>0$ and $a\neq 1$) has a maximum value of $\frac{5}{4}$ on the interval $\left[\frac{1}{4},4\right]$. Find the value of $a$.

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