
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
Calculate the average commuting time for a population, which is defined as the average time members of the group spend traveling from their homes to their workplaces in a single day. Commuters in a certain area $S$ only travel by either driving or public transportation. Analysis shows that when $x\%$ ($0 < x < 100$) of the members in $S$ drive, the average commuting time for the driving group is:
$$
f(x) = \begin{cases}
30, & 0 < x \leq 30 \\
2x + \frac{1800}{x} - 90, & 30 < x < 100
\end{cases} \text{ (in minutes)}
$$
The average commuting time for the public transportation group remains constant at $40$ minutes irrespective of $x$. Based on the analysis above, answer the following questions:
1. For what range of $x$ is the public transportation group's average commuting time less than the driving group's average commuting time?
2. Find the expression for the average commuting time $g(x)$ for the overall commuter population $S$, discuss the monotonicity of $g(x)$, and explain its practical significance.

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