
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
Given a sequence $\{a_n\}$ that satisfies $a_1=5$, $a_2=5$, and $a_{n+1}=a_n+6a_{n-1}$ (where $n\geq2$, $n\in\mathbb{N}^*$),  
(1) Find all values of $\lambda$ that make the sequence $\{a_{n+1}+\lambda a_n\}$ a geometric sequence, and explain the reason.  
(2) Find the general formula for the sequence $\{a_n\}$.

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