
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
When a train speeds up, it can increase the railway transport capacity. During operation, two successive trains must maintain a "safe interval distance" $d$ (in kilometers) which is directly proportional to the square of the train's speed $v$ (in kilometers per hour), with a proportionality constant $k = \frac{1}{4000}$. Assuming all trains have a length $l$ of $0.4$ kilometers and a maximum speed of $v_0$ (in kilometers per hour), at what speed $v$ should the train travel to maximize the unit time flow $Q = \frac{v}{l+d}$?

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