
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 $f(x)=|\log _{2}x+ax+b|$ where $(a > 0)$. Let $M_{t}(a,b)$ denote the maximum value of the function $y=f(x)$ on the interval $\left[t,t+1\right]$ where $(t > 0)$. If for any $b\in R$, $M_{t}(a,b)\geqslant a+1$, then the maximum value of the real number $t$ is ______.

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