
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
Let \( a, b, c \) be the sides of triangle \( \triangle ABC \) with \( a \leq b \leq c \), and \( R \) and \( r \) be the circumradius and inradius of \( \triangle ABC \), respectively. Define \( f = a + b - 2R - 2r \). Determine the sign of \( f \) using the measure of angle \( \angle C \).

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