
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
We call a positive integer balanced if
- it has six digits,
- each of its six digits is non-zero, and
- the product of its first three digits is equal to the product of its last three digits.
For example, 241181 is balanced since no digit equals zero and $2 \times 4 \times 1=1 \times 8 \times 1$.
(a) Determine, with justification, all balanced positive integers of the form $3 b 8 d 5 f$.
(b) Determine, with justification, a three-digit positive integer of the form $4 b c$ for which there are exactly three balanced positive integers of the form $4 b c def$.
(c) For each of $k=4,5,6,7,8,9,10$, either
- determine, with justification, an integer $ab c$ for which there are exactly $k$ balanced positive integers of the form $abc def$, or
- justify why there does not exist an integer $abc$ for which there are exactly $k$ balanced positive integers of the form $abc def$.

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