
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
Given an integer  $N \geq 4$ , determine the largest value the sum  $$ \sum_{i=1}^{\left \lfloor{\frac{k}{2}}\right \rfloor+1}\left( \left \lfloor{\frac{n_i}{2}}\right \rfloor+1\right) $$  may achieve, where  $k, n_1, \ldots, n_k$  run through the integers subject to  $k \geq 3$ ,  $n_1 \geq \ldots\geq n_k\geq 1$  and  $n_1 + \ldots + n_k = N$ .

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