
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 Joy has $40$ thin rods, one each of every integer length from $1 \text{ cm}$ through $40 \text{ cm}$, and she wants to choose a fourth rod with lengths $5 \text{ cm}$, $10 \text{ cm}$, and $20 \text{ cm}$ to form a quadrilateral with positive area, how many of the remaining rods can she choose as the fourth rod?

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