
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 *base* 10 *over-expansion* of a positive integer  $N$  is an expression of the form  $N=d_k10^k+d_{k-1}10^{k-1}+\cdots+d_0 10^0$  with  $d_k\ne 0$  and  $d_i\in\{0,1,2,\dots,10\}$  for all  $i.$  For instance, the integer  $N=10$  has two base 10 over-expansions:  $10=10\cdot 10^0$  and the usual base 10 expansion  $10=1\cdot 10^1+0\cdot 10^0.$  Which positive integers have a unique base 10 over-expansion?

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