
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 a 2023-tuple of nonnegative integers \(\left(a_{1}, a_{2}, \ldots a_{2023}\right)\) is sweet if the following conditions hold:

- \(a_{1}+a_{2}+\ldots+a_{2023}=2023\),
- \(\frac{a_{1}}{2^{1}}+\frac{a_{2}}{2^{2}}+\ldots+\frac{a_{2023}}{2^{2023}} \leq 1\).

Determine the greatest positive integer \(L\) such that

\[
a_{1}+2a_{2}+\ldots+2023a_{2023} \geq L
\]

holds for every sweet 2023-tuple \(\left(a_{1}, a_{2}, \ldots, a_{2023}\right)\).

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