
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
There is number  $1$  on the blackboard initially. The first step is to erase  $1$  and write two nonnegative reals whose sum is  $1$ . Call the smaller number of the two  $L_2$ . 
For integer  $k \ge 2$ , the  ${k}$  the step is to erase a number on the blackboard arbitrarily and write two nonnegative reals whose sum is the number erased just now. Call the smallest number of the  $k+1$  on the blackboard  $L_{k+1}$ . 
Find the maximum of  $L_2+L_3+\cdots+L_{2024}$ .

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