
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
Given \( p \) and \( q \) such that \( 0 \le p \le 1 \) and \( 0 \le q \le 1 \), define a new function
\[
  H(p, q) = -3pq + 2p(1-q) + 4(1-p)q - 5(1-p)(1-q).
\]
Define \( I(p) \) as the maximum of \( H(p,q) \) over all \( q \). What is the value of \( p \) that minimizes \( I(p) \) within the interval \( 0 \le p \le 1 \)?

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