
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
Find the total number of different integer values the function
$$
f(x)=[x]+[2x]+\left[\frac{5x}{3}\right]+[3x]+[4x]
$$
takes for real numbers $x$ with \(0 \leq x \leq 100\).

Note: \([t]\) is the largest integer that does not exceed \(t\).

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