
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
Given a triangle with sides \(a \leq b \leq c\), by the law of cosines we have:

\[ 
\cos \hat{C} = \frac{a^2 + b^2 - c^2}{2ab} 
\]

If the angle \(\hat{C}\) is obtuse, \(\cos \hat{C} < 0\). Since \(2ab\) is positive, this is the same as \(a^2 + b^2 - c^2 < 0\). Therefore, for a triangle to be obtuse, the square of the longest side is greater than the sum of the squares of the other two sides. Additionally, by the triangle inequality, we know that the longest side is less than the sum of the other two sides. We can summarize this information through the inequalities

\[
a^2 + b^2 < c^2 < (a + b)^2
\]

How many obtuse triangles can be formed with positive integer sides less than 7?

Note: Consider that two triangles with the same side lengths but in different orders represent the same triangle.

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