
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
Let all possible  $2023$ -degree real polynomials:  $P(x)=x^{2023}+a_1x^{2022}+a_2x^{2021}+\cdots+a_{2022}x+a_{2023}$ ,
where  $P(0)+P(1)=0$ , and the polynomial has 2023 real roots  $r_1, r_2,\cdots r_{2023}$  [not necessarily distinct] so that  $0\leq r_1,r_2,\cdots r_{2023}\leq1$ . What is the maximum value of  $r_1 \cdot r_2 \cdots r_{2023}?$ 

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