
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
Let $T \subset \mathbb{R}$. If there exists a constant $M > 0$ such that for any $t \in T$, $|t| \leq M$, then $T$ is called a bounded set, and $M$ is called an upper bound of the set $T$.

(1) Let $A_1 = \{y | y = \frac{2^x - 1}{2^x + 1}, x \in \mathbb{R}\}$, $A_2 = \{x | \sin x > \frac{1}{2}\}$. Determine whether $A_1$ and $A_2$ are bounded sets, and explain the reasons.

(2) Given $f(x) = x^2 + u$, let $f_1(x) = f(x)$, $f_n(x) = f(f_{n-1}(x))$ ($n=2,3,\ldots$). If $m \in \mathbb{R}$, $u \in [\frac{1}{4}, +\infty)$, and $B = \{f_n(m) | n \in \mathbb{N}^*\}$ is a bounded set, find the value of $u$ and the range of $m$.

(3) Suppose $a, b, c$ are all positive numbers. Denote the minimum value among $(a-b)^2$, $(b-c)^2$, and $(c-a)^2$ as $d$. Determine whether there exists a positive number $\lambda \in (0, 1)$ that can be an upper bound of the bounded set $C = \{y | \frac{d}{a^2 + b^2 + c^2}, a, b, c \text{ are all positive numbers}\}$. If it exists, find the minimum value of $\lambda$; if not, please explain the reasons.

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