
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
Given points $A(x\_1, y\_1)$ and $B(x\_2, y\_2)$ on the graph of the quadratic function $y = x^2 + mx + n$, where $x\_1 = 1$, $x\_2 = 3$, and $y\_1 = y\_2$.
(1) Find the value of $m$; (2) If the parabola intersects the $x$-axis at only one point, find the value of $n$; (3) If $P(a, b\_1)$ and $Q(3, b\_2)$ are two points on the function's graph and $b\_1 > b\_2$, find the range of real number values for $a$.

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