
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 the quadratic function $f(x) = ax^2 + x + 1$ $(a > 0)$.

(1) Find the maximum value $M(a)$ of the function $f(x)$ in the interval $[-4, -2]$;

(2) If the equation $f(x) = 0$ has two real roots $x_1$ and $x_2$, and $\frac{x_1}{x_2} \in [\frac{1}{10}, 10]$, find the maximum value of the real number $a$.

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