
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
$$
f(x) = \begin{cases} 
1, & 0 \leq x < \frac{1}{2} \\
-1, & \frac{1}{2} \leq x < 1 \\
0, & x < 0 \text{ or } x \geq 1
\end{cases}
$$
and
$$
g(x) = \begin{cases} 
1, & 0 \leq x < 1 \\
0, & x < 0 \text{ or } x \geq 1
\end{cases},
$$
find the expression for $g(2x)$; and if $m, n \in \mathbb{Z}$ and $m \cdot g(n \cdot x) - g(x) = f(x)$, then determine the sum $m+n$.

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