
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 that the function f(x) is defined as $$f(x) = \begin{cases} x^2 - 4ax + 3, & x < 1 \\ \log_a{x} + 2a, & x \geq 1 \end{cases}$$ and satisfies $\frac{f(x_1) - f(x_2)}{x_1 - x_2} < 0$ for any $x_1 \neq x_2$, determine the range of possible values for $a$.

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