
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 that the sequence $\{a\_n\}$ is an arithmetic sequence with non-zero terms, $S\_n$ is the sum of its first $n$ terms, and $a\_n = \sqrt{S\_{2n-1}}\ (n \in \mathbb{N}^*)$, $A = -a\_1a\_2 + a\_2a\_3 - a\_3a\_4 + a\_4a\_5 - \dots + a\_{2n}a\_{2n+1}$, find $A$.

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