
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
If the length of the step with number \( k, k=1, 2, \ldots, m \) is \( l_k \) cm, then according to the condition, \( l_k \leq a \) for any \( k \). Then there exists \( k \) for which \( \frac{100s}{m} \leq l_k \leq a \). Otherwise, the sum of the lengths of all steps is less than \( s \).

If \( a < \frac{100 s}{m-1} \), then for any pair of steps with lengths \( l_p \) and \( l_q \), the condition \( l_p + l_q > l_r \) holds for any \( r \). Indeed, for any \( k \), \( l_k \leq a < \frac{100 s}{m-1} \), and for any \( p, q \),

\[ l_p + l_q = 100s - \sum_{k \neq p, q} l_k > 100s - \frac{m-2}{m-1} 100s = \frac{100s}{m-1} > a \geq l_r \]

If \( a \geq \frac{100 s}{m-1} \), the condition of the problem is not satisfied, for example, for step lengths \( l_k = \frac{100 s}{m-1} \) for \( k=1, 2, \ldots, m-2 \) and \( l_{m-1} = l_m = \frac{100 s}{2(m-1)} < a \), since \( l_{m-1} + l_m = l_1 \).

Thus, the allowable values of \( a \) form the set \( \left[ \frac{100 s}{m}; \frac{100 s}{m-1} \right) \).

In variant 1, \( s = 120, m = 150 \). Consequently, \( a \in \left[80; \frac{12000}{149} \right) \).

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