
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 the statements $p: |4x-3| \leqslant 1$ and $q: x^2 - (2a+1)x + a(a+1) \leqslant 0$, if the statement "$\neg p \Rightarrow \neg q$" is false and "$\neg q \Rightarrow \neg p$" is true, find the range of real number values for $a$.

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