
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
Maria Ivanovna, a highly skilled English teacher, tutors schoolchildren and students. The cost of one hour (60 minutes) of Maria Ivanovna's lesson is 2000 rubles (after taxes). In addition to tutoring income, she has a steady income from investments, which amounts to 30,000 rubles per month. Maria Ivanovna dedicates weekends entirely to her family and does not work on those days. She is very disciplined and carefully manages her time and income. On working days, which are exactly 20 each month, she sleeps for nine hours a day, works for some time, dedicates part of her time to her hobby - knitting, and spends the rest of her time on rest and household chores. She considers her hobby an important part of her life, so if she works $L$ hours a day, she definitely spends $2L$ hours knitting. She does not sell her knitted items but gives them to friends, acquaintances, and those in need. Additionally, Maria Ivanovna is an active philanthropist and donates money to a children's aid fund. Every working day, she sets aside exactly $k / 3$ thousand rubles for charity, if on that day she spent $k$ hours on rest and household chores. For all household expenses, hobbies, and housekeeping (excluding charity expenses), Maria Ivanovna spends 60,000 rubles a month, and she has no other expenses. Despite loving to rest and engage in household chores, she needs to work, and each working day she decides how many lessons to conduct with students.

(a) What is the maximum amount of money Maria Ivanovna can donate to charity each month?

(b) Since Maria Ivanovna is a highly skilled English teacher, the parents of her students are willing to offer her increased pay for additional lessons (beyond those she already conducts daily). Maria Ivanovna has long planned for the days when she can no longer work, so she confidently spends all her monthly income. Will Maria Ivanovna agree to their proposal? If so, what should be the pay for the overtime lessons for her to agree to conduct them? If not, explain why she would refuse such an offer.

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