
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
If \(0 \le p \le 1\) and \(0 \le q \le 1\), define \(H(p, q)\) by
\[
  H(p, q) = -3pq + 4p(1-q) + 5(1-p)q - 6(1-p)(1-q) + 2p.
\]
Define \(J(p)\) 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 \(J(p)\)?

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