
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
Express the polynomial $f(x) = 1 - x + x^2 - x^3 + \ldots + (-1)^{19}x^{19} + x^{20}$ as a polynomial $g(y) = a + a_1y + a_2y^2 + \ldots + a_{19}y^{19} + a_{20}y^{20}$, where $y = x + 1$. Find the value of $a + a_1 + \ldots + a_{20}$.

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