
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
The infinite sequence $\{a\_n\}$, with the first term being 0, satisfies the following two conditions simultaneously:
1. $|a_{n+1} - a_n| = n$;
2. $a_n \leq \frac{n-1}{2}$.
(I) Directly write out all possible values of $a_4$;
(II) Let $b_n = a_{2n}$. If $b_n < b_{n+1}$ holds for any $n \in \mathbb{N}^*$, find the general term formula for the sequence $\{b_n\}$;
(III) For a given positive integer $k$, find the maximum value of $a_1 + a_2 + \ldots + a_k$.

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