
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 the quadratic function $y=f(x)=x^{2}+bx+c$ whose graph passes through the point $(1,13)$, and the function $y=f\left(x-\frac{1}{2}\right)$ is an even function:
(1) Find the explicit expression for $f(x)$;
(2) Determine whether there exists a point on the graph of $y=f(x)$ whose x-coordinate is a positive integer and y-coordinate is a perfect square. If such a point exists, find its coordinates; if not, provide an explanation.

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