
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 on the interval \([p, q]\), if for any \( x \in [p, q] \), \( |f(x) - g(x)| \leqslant 1 \), then \( f(x) \) and \( g(x) \) are considered to be close functions on the interval \([p, q]\); otherwise, they are considered non-close functions on \([p, q]\). Given two functions \( f_{1}(x) = \log_{a}(x - 3a) \) and \( f_{2}(x) = \log_{a} \left( \frac{1}{x - a} \right) \) (with \( a > 0 \) and \( a \neq 1 \)), and the interval \([a+2, a+3]\):

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

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