
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
For two functions $f(x)$ and $g(x)$ that are meaningful in the interval $[p, q]$, if for any $x \in [p, q]$, we have $|f(x) - g(x)| \leqslant 1$, we say that $f(x)$ and $g(x)$ are close to each other in the interval $[p, q]$. Otherwise, we say they are not close in $[p, q]$. Given two functions $f_{1}(x) = \log_{a}(x - 3a)$ and $f_{2}(x) = \log_{a} \frac{1}{x - a}$ where $a > 0$ and $a \neq 1$, and a given interval $[a + 2, a + 3]$:

1. If $f_{1}(x)$ and $f_{2}(x)$ are meaningful in $[a + 2, a + 3]$, find the range of $a$.
2. Discuss whether $f_{1}(x)$ and $f_{2}(x)$ are close functions in the interval $[a + 2, a + 3]$.

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