
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 function $f(x) = |x-a|$, $g(x) = ax$, where $a \in \mathbb{R}$.
1. If the equation $f(x) = g(x)$ has two distinct solutions, find the range of the real number $a$.
2. If $a > 0$, let $F(x) = g(x)f(x)$. Try to find the maximum value of the function $y = F(x)$ in the interval $[1, 2]$.

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