
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
\[
  F(p, q) = -2pq + 3p(1-q) + 3(1-p)q - 4(1-p)(1-q).
\]
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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