
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
Determine the largest integer $M > 1$ with the following property: For any 10 distinct real numbers chosen from the interval $[1, M]$, it is always possible to select three numbers, denoted in increasing order as $a < b < c$, such that the quadratic equation $ax^2 + bx + c = 0$ has no real roots.

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