
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
Find largest possible constant  $M$  such that, for any sequence  $a_n$ ,  $n=0,1,2,...$  of real numbers, that satisfies the conditions :
i)  $a_0=1$ ,  $a_1=3$ ii)  $a_0+a_1+...+a_{n-1} \ge 3 a_n - a_{n+1}$  for any integer  $n\ge 1$ to be true that  $$ \frac{a_{n+1}}{a_n} >M $$   for any integer  $n\ge 0$ .

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