
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
Here is the algorithm for calculating the value of the polynomial $f(x) = 1 + x + 2x^2 + 3x^3 + 4x^4 + 5x^5$ at $x = -1$ using the Horner's method:  
$a_5 = 5$, $u = a_5 = 5$;  
$a_4 = 4$, $u_1 = ux + a_4 = -5 + 4 = -1$;  
$a_3 = 3$, $u_2 = u_1x + a_3 = 1 + 3 = 4$;  
$a_2 = 2$;  
$a_1 = 1$, $u_4 = u_3x + a_1 = 2 + 1 = 3$;  
$a = 1$, $u_5 = u_4x + a = -3 + 1 = -2$;  
Therefore, $f(-1) = \boxed{-2}$.

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