
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 young man, returning home from vacation on his bicycle, traveled 246 km and spent on this journey one day more than half of the remaining days until the end of his vacation. Now, he has two options to cover the remaining 276 km to arrive home exactly on time: either travel $h$ km more daily than initially planned, or maintain the original daily distance and exceed it by $2h$ km only on the last day of the journey. How many days before the end of his vacation did he start heading home, given that the number of days is an integer?

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