
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
Using the digits \(\mathrm{a}\), \(\mathrm{b}\), and \(\mathrm{c}\), we construct a three-digit number \(\mathrm{a b c}\) and the two-digit numbers \(\mathrm{a b}\), \(\mathrm{b c}\), and \(\mathrm{c a}\). Find all possible values of \(\mathrm{a}\), \(\mathrm{b}\), and \(\mathrm{c}\) such that \(\frac{\mathrm{abc}+\mathrm{a}+\mathrm{b}+\mathrm{c}}{\mathrm{ab}+\mathrm{bc}+\mathrm{ca}}\) is an integer.

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