
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 equation of a circle with center $(a,b)$ is $(x-a)^{2}+(y-b)^{2}=1$ where $a \in \mathbb{R}$ and $b \in \mathbb{R}$, and the equation of another circle symmetrical to it with respect to the line $y=x+1$ is $(x-1)^{2}+(y-3)^{2}=1$, determine the value of $a+b$.

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