
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
**p1.** You have bought a box which contains six unsharpened pencils. Is it possible to arrange them so that every pair of pencils touch.**p2.** Is it possible to put  $54$  rabbits in  $10$  cages so that every pair of cages have a different number of rabbits, and each cage contains at least one rabbit?**p3.** The year  $1983$  had  $53$  Saturdays. What day of the week was January  $1$ ,  $1983$ ?**p4.** Suppose  you play the following game with a friend: Your friend picks a whole number between  $1$  and  $100$ . You choose a number and ask your friend to compare it with his number. He tells whether your number is bigger, smaller , or equal to his. Tell how you can always guess his number in no more than seven tries.


PS. You should use hide for answers.

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