
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

Once upon a time, 45 friends living in different corners of the globe wanted to share news with each other. To do this, they decided to hold $k$ video meetings, where each person would share their news with everyone, as well as all the news from others that they had learned previously.

Ten days were proposed for the video meetings, but it turned out that each of the friends could only be present on eight of those days. What is the smallest natural number $k$ such that $k$ days can be selected from the proposed 10 days for the video meetings, so that everyone will learn the news from each other?

(Note: No new news arises between the proposed days, and they do not communicate with each other in any other way. One video meeting is held on each of the proposed days, attended by all who can be present on that day.)

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