
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 \( 0 \leq p \leq 1 \) and \( 0 \leq q \leq 1 \), define \( H(p, q) \) by
\[
  H(p, q) = -3pq + 4p(1-q) + 4(1-p)q - 5(1-p)(1-q).
\]
Define \( J(p) \) to be the maximum of \( H(p, q) \) over all \( q \) (in the interval \( 0 \leq q \leq 1 \)). What is the value of \( p \) (in the interval \( 0 \leq p \leq 1 \)) that minimizes \( J(p) \)?

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