
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

Maxim Andreevich, a former university professor, teaches mathematics to groups of schoolchildren. The cost for one hour (60 min) of Maxim Andreevich's lessons with a group of schoolchildren is 3 thousand rubles (after taxes). In addition to tutoring income, he has a steady income from investment activities, which amounts to 14 thousand rubles per month. He fully dedicates his weekends to his family and does not work. Maxim Andreevich is very disciplined and strictly monitors his time and income distribution. On working days, which number exactly 21 each month, he sleeps eight hours a day, works some hours, devotes some time to his hobby of creating metal engravings, and spends the remaining time resting and managing household chores. He considers his hobby an important part of his life, so if he works L hours a day, he definitely spends 2L hours on creating engravings. He does not sell these engravings but gives them to friends and acquaintances. Additionally, Maxim Andreevich is an active philanthropist and donates money to a children's charity fund. Each working day he sets aside exactly \( \frac{k}{3} \) thousand rubles for charity if he spent exactly \( k \) hours on rest and household chores that day. For all household expenses, hobbies, and running the household (except for charity expenses), Maxim Andreevich spends 70 thousand rubles per month, and he has no other expenses. Of course, Maxim Andreevich loves to rest and handle household chores, but he has to work, and each working day he decides how many lessons he will conduct with students.

(a) What is the maximum amount of money Maxim Andreevich can donate monthly to charity?

(b) Since Maxim Andreevich is an excellent math teacher, the parents of the students he teaches are willing to offer him higher pay for additional lessons (beyond those he already gives daily). Maxim Andreevich has long taken care of the days when he can no longer work, so he confidently spends all the money he receives monthly. Will Maxim Andreevich agree to their offer? If yes, what should be the payment for overtime lessons so that he agrees to conduct additional lessons? If no, explain why he would refuse such an offer.

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