
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
In right triangle ABC, the lengths of the three sides are a, b, and c, with c being the hypotenuse.
(1) If a, b, c are in geometric progression and a=2, find the value of c;
(2) Assuming a, b, c are positive integers.
    (i) If a, b, c are three consecutive integers, find the area of triangle ABC;
    (ii) If a, b, c are in arithmetic progression, list the areas of these triangles in ascending order as a sequence where the nth term is denoted by $S_n$, and let $T_n = S_1 - S_2 + S_3 - \ldots + (-1)^n S_n$, find all values of n for which $|T_n| > 3 \cdot 2^n$.

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