
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
Masha and Misha left their homes simultaneously, walking towards each other, and met 1 kilometer away from Masha's home. On another occasion, they left their homes simultaneously again, but this time Masha walked at twice her original speed, and Misha walked at half his original speed. They met 1 kilometer away from Misha's home this time. What is the distance between Masha's and Misha's homes?

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