
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 \(F(p, q)\) by
\[
F(p, q) = -4pq + 5p(1-q) + 2(1-p)q - 3(1-p)(1-q).
\]
Define \(G(p)\) to be the maximum of \(F(p, q)\) 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 \(G(p)\)?

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