
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
For a function $y=f(x)$ with $x \in D$, if for any $x_1 \in D$ there exists a unique $x_2 \in D$ such that $\sqrt{f(x_1)f(x_2)}=M$, the function $f(x)$ is said to have a geometric mean of $M$ on $D$. Find the geometric mean $M$ of the function $f(x) = x^3 - x^2 + 1$ on the interval $x \in [1,2]$.

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