
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
Use the Horner's method to calculate the value of $f(x) = 8x^7 + 5x^6 + 3x^4 + 2x + 1$ at $x = 2$ and $x = -1$, and determine whether the polynomial $f(x)$ has a zero in the interval $[-1, 2]$.

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