
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

Construct the equation of the common tangent to the graphs of the functions \(y = 1 + x - x^2\) and \(y = 0.5(x^2 + 3)\). Let \(y = ax + b\) be the equation of the common tangent. Then:

\[1 + x - x^2 = ax + b, \quad x^2 + (a-1)x + b - 1 = 0\]

\[D = (a-1)^2 - 4b + 4 = 0, \quad a^2 - 2a - 4b + 5 = 0\]

\[0.5(x^2 + 3) = ax + b, \quad x^2 - 2ax - 2b + 3 = 0\]

\[\frac{D}{4} = a^2 - (-2b + 3) = 0, \quad a^2 + 2b - 3 = 0\]

From \((*)\) and \((**)\) we get, \(3a^2 - 2a - 1 = 0, \quad a_1 = 1, \quad a_2 = -1/3\)

\[b = \frac{3 - a^2}{2}, \quad b_1 = 1, \quad b_2 = \frac{13}{9}\]

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