
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 the sequence $\left\{a_{n}\right\}$, given $a_{1}=2$, for any $p, q \in \mathbf{Z}_{+}$, it holds that
$$
a_{p+q}=a_{p}+a_{q}.
$$
(1) Find the explicit formula for the sequence $\left\{a_{n}\right\}$.
(2) If the sequence $\left\{b_{n}\right\}$ satisfies
$$
\begin{aligned}
a_{n}= & \frac{b_{1}}{2+1}-\frac{b_{2}}{2^{2}+1}+\frac{b_{3}}{2^{3}+1}-\frac{b_{4}}{2^{4}+1}+\cdots+ \\
& (-1)^{n-1} \frac{b_{n}}{2^{n}+1}\left(n \in \mathbf{Z}_{+}\right),
\end{aligned}
$$
find the explicit formula for the sequence $\left\{b_{n}\right\}$.
(3) Let $c_{n}=3^{n}+\lambda b_{n}\left(n \in \mathbf{Z}_{+}\right)$. Is there a real number $\lambda$ such that $c_{n+1}>c_{n}$ always holds for all $n \in \mathbf{Z}_{+}$? If it exists, determine the range of real number $\lambda$; if it does not exist, provide an explanation.

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