
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
To compute the value of the polynomial \( P(x) = a_{n}x^{n} + a_{n-1}x^{n-1} + \ldots + a_{0} \) at \( x = x_{0} \), a straightforward method is to calculate \( a_{n}x_{0}^{n} \), \( a_{n-1}x_{0}^{n-1} \), ..., \( a_{1}x_{0} \), and then sum all the obtained values along with \( a_{0} \). This requires \( 2n - 1 \) multiplications (computing \( x_{0}^{k} \) for \( k = 2, 3, \ldots, n \) requires \( n - 1 \) multiplications). Devise a method to compute \( P(x_{0}) \) that requires only \( n \) multiplications and \( n \) additions (Horner's scheme).

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