
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
A morning passenger train from Budapest to Nové Zámky travels at a speed of \( v_{1} \). This train should be caught by a later departing express train with a speed of \( v_{2} \) before reaching Nové Zámky. After the passenger train has traveled \(\frac{2}{3}\) of its journey, it continues at half of its original speed, causing the express train to catch up with it \(d\) kilometers before reaching the Nové Zámky station. What is the distance between the two stations if \( d = 27 \frac{1}{9} \, \text{km} \), the passenger train travels \( \frac{1}{2} \, \text{km} \) per minute, and the express train travels \( 1 \, \text{km} \) per minute?

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