
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

To wash off the pesticide residue from a batch of vegetables using water, assume the following about the effectiveness of washing once with a certain amount of water: Washing with 1 unit of water can remove $\frac{1}{2}$ of the pesticide residue on the vegetables. The more water used, the more pesticide is removed, though some residue always remains. Suppose after washing once with $x$ units of water, the ratio of pesticide residue on the vegetables to the pesticide residue before washing is the function $f(x)$.
(1) Specify the value of $f(0)$ and explain its practical significance;
(2) Based on the assumptions, write the conditions that the function $f(x)$ should satisfy and its properties;
(3) Given $f(x)=\frac{1}{1+x^{2}}$, if you have $a$ units of water (where $a>0$), you can either wash once or split the water into 2 equal parts and wash twice. Which method removes more pesticide residue? Explain the reasoning.

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