
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
Define a new function $H(p, q)$ for $0 \le p \le 1$ and $0 \le q \le 1$ as follows:
\[
  H(p, q) = -3pq + 4p(1-q) + 4(1-p)q - 6(1-p)(1-q).
\]
Define $K(p)$ to be the maximum of $H(p, q)$ over all $q$ (in the interval $0 \le q \le 1$). What is the value of $p$ (in the interval $0 \le p \le 1$) that minimizes $K(p)$?

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