
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
For a positive integer \( x \), let \( m = \left\lfloor \frac{x}{2} \right\rfloor + \left\lfloor \frac{x}{2^2} \right\rfloor + \left\lfloor \frac{x}{2^3} \right\rfloor + \cdots + \left\lfloor \frac{x}{2^k} \right\rfloor \), where \( k \) is the smallest integer such that \( 2^k \geq x \) and the symbol \(\left\lfloor x \right\rfloor\) denotes the greatest integer less than or equal to \( x \). The difference \( x - m \), referred to as the "loss number" of the positive integer \( x \), is given. For example, when \( x = 100 \),

\[ \begin{aligned}
m & = \left\lfloor \frac{100}{2} \right\rfloor + \left\lfloor \frac{100}{2^2} \right\rfloor + \left\lfloor \frac{100}{2^3} \right\rfloor + \left\lfloor \frac{100}{2^4} \right\rfloor + \left\lfloor \frac{100}{2^5} \right\rfloor + \left\lfloor \frac{100}{2^6} \right\rfloor + \left\lfloor \frac{100}{2^7} \right\rfloor \\
& = 50 + 25 + 12 + 6 + 3 + 1 + 0 = 97,
\end{aligned} \]

\[ x - m = 100 - 97 = 3, \]

so the "loss number" of \( 100 \) is \( 3 \). Find the smallest positive integer \( x \) for which the "loss number" is \( 9 \).

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