
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
Two people, A and B, simultaneously solve the radical equation \(\sqrt{x+a}+\sqrt{x+b}=7\). When copying the problem, A miswrites it as \(\sqrt{x-a}+\sqrt{x+b}=7\) and finds one of the solutions is \(12\). B miswrites it as \(\sqrt{x+a}+\sqrt{x+d}=7\) and finds one of the solutions is \(13\). Given that both individuals solve their respective miswritten problems correctly and \(a\), \(b\), and \(d\) are integers, find the values of \(a\) and \(b\).

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