
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 polynomial  $f$  with integer coefficients is written on the blackboard. The teacher is a mathematician who has  $3$  kids: Andrew, Beth and Charles. Andrew, who is  $7$ , is the youngest, and Charles is the oldest. When evaluating the polynomial on his kids' ages he obtains:
[list] $f(7) = 77$  $f(b) = 85$ , where  $b$  is Beth's age, $f(c) = 0$ , where  $c$  is Charles' age.[/list]
How old is each child?

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