
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

A random point \(N\) on a line has coordinates \((t, -2-t)\), where \(t \in \mathbb{R}\). A random point \(M\) on a parabola has coordinates \(\left( x, x^2 - 4x + 5 \right)\), where \(x \in \mathbb{R}\). The square of the distance between points \(M\) and \(N\) is given by \(\rho^2(x, t) = (x - t)^2 + \left( x^2 - 4x + 7 + t \right)^2\). Find the coordinates of points \(M\) and \(N\) that minimize \(\rho^2\).

When the point \(M\) is fixed, \(\rho^2(t)\) depends on \(t\), and at the point of minimum, its derivative with respect to \(t\) is zero: \[-2(x - t) + 2\left( x^2 - 4x + 7 + t \right) = 0.\]

When point \(N\) is fixed, the function \(\rho^2(x)\) depends on \(x\), and at the point of minimum, its derivative with respect to \(x\) is zero:
\[2(x - t) + 2\left( x^2 - 4x + 7 + t \right)(2x - 4) = 0.\]

We solve the system:
\[
\begin{cases}
2 \left( x^2 - 4x + 7 + t \right) (2x - 3) = 0 \\
4(x - t) + 2 \left( x^2 - 4x + 7 + t \right) (2x - 5) = 0
\end{cases}
\]

**Case 1: \(x = \frac{3}{2}\)**

Substituting \(x = \frac{3}{2}\) into the second equation, we get \(t = -\frac{7}{8}\). Critical points are \(N^* \left( -\frac{7}{8}, -\frac{9}{8} \right)\) and \(M^* \left( \frac{3}{2}, \frac{5}{4} \right)\). The distance between points \(M^*\) and \(N^*\) is \(\frac{19\sqrt{2}}{8}\). If \(2r \leq \frac{19\sqrt{2}}{8}\), then the circle does not intersect the parabola (it touches at points \(M^*\) and \(N^*\)).

**Case 2: \(x \neq \frac{3}{2}\)**

Then \(\left( x^2 - 4x + 7 + t \right) = 0\) and from the second equation \(x = t\). Substituting \(x = t\) into the equation \(\left( x^2 - 4x + 7 + t \right) = 0\), we get the condition for \(t\), i.e. \(t^2 - 3t + 7 = 0\). Since this equation has no solutions, case 2 is not realizable. The maximum radius \(r_{\max} = \frac{19\sqrt{2}}{16}\).

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