
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 that Joy has $30$ thin rods, one each of every integer length from $1 \text{ cm}$ through $30 \text{ cm}$, and she places the rods with lengths $5 \text{ cm}$, $10 \text{ cm}$, and $20 \text{ cm}$ on a table, determine how many of the remaining rods she can choose as the fourth rod to form a quadrilateral with a positive area.

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