
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
If the expression of a function equals the sum of the expressions of two other functions, then this function is called the "generating function" of the other two functions. There are two quadratic functions about $x$, $y_1$, $y_2$, and $y_1 = a(x-m)^2 + 4$ ($m > 0$), the "generating function" of $y_1$, $y_2$ is: $y = x^2 + 4x + 14$; when $x = m$, $y_2 = 15$; the vertex of the graph of the quadratic function $y_2$ is $(2, k)$.  
(1) Find the value of $m$;  
(2) Find the expressions of the quadratic functions $y_1$, $y_2$.

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