
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 the greatest positive real number  $M$  such that for all positive real sequence  $(a_n)$  and for all real number  $m < M$ , it is possible to find some index  $n \ge 1$  that satisfies the inequality  $a_1 + a_2 + a_3 + ...+ a_n +a_{n+1} > m a_n$ .

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