
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 that the center of circle $C$ lies on the line $y=x+1$, with a radius of $\sqrt {2}$, and the circle passes through point $P(5,4)$:
1. Find the standard equation of circle $C$;
2. Find the equation of the tangent line passing through point $A(1,0)$ and tangent to circle $C$.

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