
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 a sequence $\{a\_n\}$ where each term is positive, with the first term $a\_1=1$. $S\_n$ represents the sum of the first $n$ terms of the sequence, and it satisfies the equation: $a\_nS_{n+1}-a_{n+1}S_n+a_n-a_{n+1}=λa_na_{n+1} (λ≠0, n∈N^∗)$.

(1) If $a\_1$, $a\_2$, $a\_3$ form a geometric sequence, find the value of the real number $λ$;

(2) If $λ= \frac{1}{2}$, find $S\_n$.

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