
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
In her fourth year at Hogwarts, Hermione felt outraged by the treatment of house elves and created the Society for the Promotion of Elfish Welfare (S.P.E.W.). Naturally, even the most noble and enlightened idea requires funds for promotion, so Hermione decided she could finance her campaign by producing and selling badges. Everyone at Hogwarts liked this idea, and Hermione secured the right to manufacture these badges by obtaining a corresponding patent. 

Ron, who had known the intricacies of magical technology since childhood, convinced Hermione that the production costs of the badges would be $T C(Q) = 5 Q^2$, where $Q$ is the number of badges produced. Harry, with the help of his friends and Polyjuice Potion, determined that the demand for badges from all houses except Slytherin is represented by the demand function $q_{ws} = 26 - 2P$, while the demand from Slytherin is represented by the demand function $q_s = 10 - P$ (Slytherin students are biased against ideas from Gryffindors, especially those of Muggle origin, but they would not mind supporting the society).

What is the maximum amount Hermione can earn to support her campaign from selling the badges if the prices of the same item must be identical for all houses at Hogwarts?

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