
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
A sequence $\{a_n\}$ satisfies $a_1=2$, $a_{n+1}=a_n+\lambda\cdot 2^n$, and $a_1, a_2 + 1, a_3$ form an arithmetic sequence, where $n\in\mathbb{N}^+$.

(1) Find the value of the real number $\lambda$ and the general formula for the sequence $\{a_n\}$.
   
(2) If the inequality $\dfrac{p}{2n-5} \leqslant \dfrac{2p+16}{a_n}$ holds for exactly four natural numbers $n$, find the value of the positive integer $p$.

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