
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 $a_n$ satisfying $a_1=1$, and for all $n \in \mathbb{N}^*$, the recursive relation is given by 
$$
4a_{n+1} - a_n a_{n+1} + 2a_n = 9.
$$
(1) Find the values of $a_1, a_2, a_3, a_4$.  
(2) Conjecture a general formula for $a_n$ and provide a proof.

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