
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
A point $P$ is taken on the line $y=x+1$ to draw two tangents $l_{1}$ and $l_{2}$ to the circle $C:(x-1)^{2}+(y-6)^{2}=2$. When the lines $l_{1}$ and $l_{2}$ are symmetric about the line $y=x+1$, determine the length of $PC$.

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