
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
We say that a  $2023$ -tuple of nonnegative integers  $(a_1,\hdots,a_{2023})$  is *sweet* if the following conditions hold:



- $a_1+\hdots+a_{2023}=2023$
- $\frac{a_1}{2}+\frac{a_2}{2^2}+\hdots+\frac{a_{2023}}{2^{2023}}\le 1$


Determine the greatest positive integer  $L$  so that \[a_1+2a_2+\hdots+2023a_{2023}\ge L\] holds for every sweet  $2023$ -tuple  $(a_1,\hdots,a_{2023})$ *Ivan Novak*



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