
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 each positive integer $x$, let $h(x)$ denote the greatest power of $2$ that divides $x$. For example, $h(3) = 1$ and $h(12) = 4$. For each positive integer $m$, define $T_m = \prod_{j=1}^{2^m} h(j)$. Find the smallest integer $m > 0$ such that $T_m$ is divisible by $2^{1000}$.

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