
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 functions $f(x)=|x-a|$ and $g(x)=ax$ where $a \in \mathbb{R}$.

(1) If $a=1$, solve the equation $f(x)=g(x)$.

(2) If the equation $f(x)=g(x)$ has two solutions, find the range of values for the real number $a$.

(3) 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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