
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
For the sequence $\{a_{n}\}$, let $A_{n}=a_{1}+2a_{2}+\ldots +2^{n-1}a_{n}$ be the "weighted sum" of the sequence $\{a_{n}\}$. It is known that the "weighted sum" $A_{n}=n\cdot 2^{n+1}$. Let the sum of the first $n$ terms of the sequence $\{a_{n}+pn\}$ be $T_{n}$. If $T_{n}\leqslant T_{6}$ holds for any $n\in \mathbb{N}^{*}$, then the range of real number $p$ is ____.

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