
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
The polynomial \( p_1 = x - a \) may have a root \( x = a \) coinciding with one of the roots of the product \( p(x) = p_1(x) \cdot p_2(x) \).

Case 1: \( a = 1 \)

Then the polynomial \( p_2(x) = (x-1)^r(x-2)^s(x+3)^t \), where \( r \geq 1, s \geq 1, t \geq 1 \) are integers, and \( r+s+t=4 \), satisfies the condition. The polynomial sum \( p_1(x) + p_2(x) \) has the constant term \( u = (-1)^{r+s} \cdot 2^s \cdot 3^t - 1 = (-1)^t \cdot 2^s \cdot 3^t - 1 \) which has a maximum value of \( u_{\max} = 17 \) for \( r = 1, s = 1, t = 2 \).

Case 2: \( a = 2 \)

Then the polynomial \( p_2(x) = (x-1)^r(x-2)^s(x+3)^t \) and the polynomial \( p_1(x) + p_2(x) \) has the constant term \( u = (-1)^{r+s} \cdot 2^s \cdot 3^t - 2 = (-1)^t \cdot 2^s \cdot 3^t - 2 \), which has a maximum value of \( u_{\max} = 16 \) for \( r = 1, s = 1, t = 2 \).

Case 3: \( a = -3 \)

The constant term \( u = (-1)^{r+s} \cdot 2^s \cdot 3^t + 3 \) has a maximum value of \( u_{\max} = 21 \) for \( s = r = 1, t = 2 \).

The desired polynomials are: \( p_2(x) = (x-1)(x-2)(x+3)^2 \) and \( p_1(x) = x + 3 \).

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