
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
"*City Sea View*" refers to the phenomenon of urban waterlogging that has been experienced in many large and medium-sized cities in recent years. Such incidents are caused by a combination of factors, including weather conditions and urban planning, but an important contributing factor is also debris and waste blocking the drainage systems. During heavy rain, the urban waste is flushed into the sewers; and it has been noted that, without other concerns, the drainage volume $V$ (in cubic meters per hour) of a certain sewer section is a function of the density of waste and debris $x$ (in kilograms per cubic meter). When the density of waste in the sewers reaches 2 kg/m³, it leads to blockages and the drainage volume drops to 0; whereas, when the density of waste does not exceed 0.2 kg/m³, the drainage volume is 90 m³/h. Research indicates that when $0.2 \leq x \leq 2$, the drainage volume $V$ is a linear function of the waste density $x$.  
(1) Determine the expression for the function $V(x)$ when $0 \leq x \leq 2$;  
(2) At what waste density $x$ can the volume of waste (the amount of waste that passes through a section of the sewer in a unit of time, measured in kilograms per hour) $f(x) = x \cdot V(x)$ reach its maximum, and calculate this maximum value.

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