
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 say that a positive integer \( n \) is \( m \)-expressible if one can write an expression evaluating to \( n \) in base 10, where the expression consists only of:
- Exactly \( m \) numbers from the set \(\{0, 1, \ldots, 9\}\)
- The six operations \(+,-,\times,\div\), exponentiation \(\wedge\), and concatenation \(\oplus\)
- Some number (possibly zero) of left and right parentheses.

For example, 5625 is 3-expressible (in two ways), as \( 5625 = 5 \oplus (5^{\wedge} 4) = (7 \oplus 5)^{\wedge} 2 \).

Does there exist a positive integer \( A \) such that all positive integers with \( A \) digits are \((A-1)\)-expressible?
(Krit Boonsiriseth)

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