
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 a circle $C$ with a radius of $\sqrt {5}$, whose center lies on the line $l_{1}$: $x-y+1=0$, and intersects with line $l_{2}$: $\sqrt {3}x-y+1-\sqrt {3}=0$ at points $M$ and $N$ with $|MN|= \sqrt {17}$.

(1) Find the standard equation of circle $C$.

(2) When the coordinates of the center of circle $C$ are both integers, find the range of real number $a$ such that for any $m \in \mathbb{R}$, line $l_{3}$: $mx-y+\sqrt {a}+1=0$ always intersects with circle $C$.

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