
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
(1) Factorize the following expressions and write the results directly on the line:  
$x^{2}-6x+9=$ ______ , $25x^{2}+10x+1=$ ______ , $4x^{2}+12x+9=$ ______ .  
(2) Observing the coefficients of the above three polynomials, we have $(-6)^{2}=4 \times 1 \times 9$, $10^{2}=4 \times 25 \times 1$, $12^{2}=4 \times 4 \times 9$. Thus, Xiao Ming guesses: If the polynomial $ax^{2}+bx+c(a > 0)$ is a perfect square, then there must exist some relationship between the coefficients $a$, $b$, and $c$. Please express Xiao Ming's conjecture in a mathematical formula. ______ (Note: If you cannot guess the result, please write another perfect square different from those in (1) and describe the relationship among its coefficients.)  
(3) If the polynomials $x^{2}+ax+c$ and $x^{2}+cx+a$ are both perfect squares, use the rule from (2) to find the value of $ac$.

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